The Proca equations (1936) are used in quantum field theory to describe vector bosons (spin 1) with a nonzero mass. The Proca equations are not gauge invariant. In contrast to Stueckelberg's approach (1938), this article presents a gauge invariant generalization of the Proca equation by introducing an additional vector field into the Proca equation. The results are extended to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

In this paper, we explain how classical $W$-algebras can be obtained as limits of $q$-deformed $W$-algebras in type $A$. To this end, we give a detailed examination of the construction of deformed $W$-algebras via Hamiltonian reduction of deformed affine Poisson algebras. We first derive an affine Poisson vertex algebra (PVA) from the Poisson structure of the deformed affine Poisson algebra. This construction identifies a specific set of generators of the deformed $W$-algebra with generators of the classical $W$-algebra.

The process of measuring quantum observables has been plagued, since the inception of quantum mechanics, by the so-called measurement problem: it is impossible to read a definite outcome on a quantum scale. In this paper, we overcome this century-old problem by proving the existence of a measurement scheme in which the probe undergoes a quantum-to-classical transition, all the while satisfying the requirements of measurements in standard quantum mechanics.

We give a solution to Polchinski's equation for the Wilsonian effective action in terms of an expansion in binary trees, and prove that this expansion converges in fermionic field theories, provided the fermionic covariance has finite determinant and decay constants. A novel element of the proof are detailed combinatorial estimates for the number of leaf trees associated to binary trees. The method can be used on standard models of fermionic quantum field theory and quantum statistical mechanics.

We study the resurgent structure of short-time heat kernel asymptotics from the viewpoint of Picard-Lefschetz theory. For a real analytic Riemannian manifold, we show the heat kernel admits a 1-Gevrey small-time expansion whose Borel transform detects complex-geometric data beyond the real geodesic sector. We formulate an infinite-dimensional Picard-Lefschetz problem of Morse-Floer type for the holomorphic energy functional on the complexified path space, and propose a heat-kernel analogue of the Picard-Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat-kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. We perform a confirming test of this proposal on the hyperbolic plane $H^2$.

We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form thus is not immediate.

This paper presents a complete two-dimensional theoretical model for the electromagnetic behavior of square-section solid magnetic bars under sinusoidal loading. Through the application of Maxwell's equations within a Cartesian coordinate system and the integration of complex permeability, exact mathematical expressions are derived for mutual impedance, internal magnetic fields, flux, and core losses. Hyperbolic functions are utilized to separate the variables, enabling the accurate representation of edge flux accumulation and the 2D skin effect. In addition to mathematically decoupling eddy current and hysteresis losses, this formulation yields a new apparent permeability parameter. This parameter establishes a fast, reliable method for magnetic steel characterization that bypasses the extensive processing times associated with Finite Element Analysis (FEA). Numerical results over 1 Hz-1 MHz show the apparent relative permeability decreasing from 500 to 300 and a characteristic resistance peak near 700 kHz, marking the transition from volumetric to surface-dominated loss regimes.

We consider Gaussian free field (GFF) on simply connected domains with piecewise constant Dirichlet boundary data. We show that the crossing probabilities for its level lines are determined by conformal blocks of primary fields in a conformal field theory (CFT) with central charge $c = 1$ which are degenerate at each insertion. Alternatively, the crossing probabilities are ratios of explicit partition functions of fused multiple $\mathrm{SLE}_4$ curves, which can be written in terms of fused Specht polynomials introduced recently by Lafay, Peltola, & Roussillon in a representation-theoretic context. We also prove that for the metric graph GFF introduced by Lupu, with appropriate boundary conditions, the crossing probabilities for its level sets converge in the scaling limit to our formulas. In particular, the geometry of the level-set percolation for both the continuum GFF and the metric graph GFF has a CFT description in terms of the aforementioned $c=1$ conformal blocks, which are linearly independent and solve the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs of arbitrary orders. Interestingly, not all combinatorial boundary conditions are amenable for the GFF models -- while the CFT contains conformal blocks of primary fields labeled by generalized Dyck paths (i.e., semi-standard Young tableaux), the ones appearing in the above models satisfy specific monotonicity constraints.

We study a symmetric quartic deformation of the classical ($k=1$) Tempesta-Turbiner-Winternitz system, $H=\frac{1}{2}\left(P_X^2+P_Y^2\right)+X^2+Y^2 +\frac{\gamma}{X^2}+\frac{\gamma}{Y^2} +\kappa \,X^2Y^2 $, which interpolates between the maximally superintegrable Smorodinsky-Winternitz oscillator and the quartically coupled Contopoulos oscillator. For $\gamma>0$, the inverse-square terms generate impenetrable walls at ($X=0$) and ($Y=0$), splitting configuration space into invariant sectors. We show that, for sufficiently small nonzero $\kappa$, resonant averaging produces a phase-locked periodic orbit whose fixed-energy reduced Poincaré map has no unit characteristic multiplier. The Poincaré's nonintegrability criterion then excludes a second independent $C^1$ first integral in any invariant neighbourhood of this orbit. The result is local in phase space and perturbative in $\kappa$. At finite coupling, Poincaré sections and finite-time Lyapunov maps show the breakup of invariant curves and the growth of chaotic layers with increasing energy and quartic coupling. Comparison with the barrier-free Contopoulos limit shows that the singular TTW walls suppress phase-space transport without restoring integrability.

We consider localized oscillation of an Euler--Bernoulli beam on a visco-elastic foundation coupled to a damped discrete oscillator. All parameters of the system independently vary in time in a slow manner. For the conservative case, we use three various analytic approaches. Namely, these are asymptotics, the method based on the adiabatic invariance of the action of a trapped wave, and the consideration of the equivalent Hamiltonian system. All approaches result in the same formula for the amplitude of oscillation. In the dissipative case, we obtain the amplitude of oscillation only utilizing the asymptotic approach.

We formulate a finite-proper-time version of QED defined by a gauge-covariant generating functional in which every Schwinger proper-time integral has a physical lower endpoint $s_0=\Lambda^{-2}$ that is not removed. Closed fermion loops are defined by the gauge-covariant heat-kernel determinant, and open fermion lines are obtained by functional differentiation of the same finite-proper-time open-line kernel. The Ward--Takahashi identity follows exactly from gauge covariance to all orders. The vacuum polarisation is finite and exactly transverse. The on-shell mass relation gives 0.482$$ MeV for an illustrative choice $\Lambda=13\$ TeV at one loop. The free finite-proper-time propagators are proved to satisfy Osterwalder--Schrader reflection positivity, with a positive Kallen--Lehmann spectral function and no additional ghost states. The on-shell residue $Z_2>0$ at one loop confirms unitarity of the fermion sector, and the perturbative spectral function of the dressed propagator is positive. A worldline bound $Q\le0$ gives fixed-order Euclidean UV finiteness. The one-loop vertex correction gives a finite anomalous magnetic moment, a calculable $O(m^2/\Lambda^2)$ prediction absent in standard QED.

In quantum field theory on curved spacetime, and in locally-covariant quantum field theory, the Hadamard condition is often presented as a necessary condition on 'physically reasonable' states of the quantum field, and plays a central role in many theoretical and foundational applications - ranging from proofs of the renormalizability of Wick polynomials to derivations of the Hawking temperature. Yet despite this, the philosophical and foundational underpinnings of the Hadamard condition remain murky. I critically discuss existing motivations for the Hadamard condition in the literature, before arguing in favour of an alternative justification for the Hadamard condition, according to which it is best understood as a necessary and sufficient condition for the existence of a well-defined operator product on a sufficiently large space of observables of the quantum field, satisfying a variety of further conditions (thus proving a converse to a result which was already discussed in this context). This clarifies the role and status of the Hadamard condition, including its relationship to the equivalence principle, to well-definedness of physical quantities such as Wick polynomials and the expectation value of the stress-energy operator, and the sense in which Hadamard states are 'vacuum-like'.

We introduce and study compatible Lie conformal bialgebras as conformal counterparts of compatible Lie bialgebras. Such a structure consists of two compatible Lie conformal brackets and two compatible conformal cobrackets on the same $\C[\partial]$-module, and every simultaneous linear combination of them is again a Lie conformal bialgebra. We develop representations and matched pairs for compatible Lie conformal algebras, introduce compatible Lie conformal coalgebras, and establish their duality through the conformal dual in the finite case. For $\C[\p]$-modules that are free of finite rank, we prove the equivalence among compatible Lie conformal bialgebras, standard compatible conformal Manin triples and matched pairs. In the coboundary case, we characterize the tensors $r$ that determine compatible Lie conformal bialgebras. The characterization requires the symmetric part of $r$ to be invariant with respect to both brackets and three conformal Yang--Baxter conditions to hold. The first two conditions correspond to the two brackets separately, whereas the third is the compatible conformal Yang--Baxter condition. For comparison, we introduce the compatible conformal classical Yang--Baxter equation (CYBE) and show that each of its solutions satisfies these three conditions, while the converse fails. One explicit example shows that the first two conditions do not imply the third. Another shows that all three conformal Yang-Baxter conditions may hold even though $r$ is not a solution of compatible conformal CYBE.

In this article, we define and study the statistical physics of planar Carrollian systems. While it has been shown recently that, for general dimensions, the Carroll limit of Poincaré statistical physics typically does not converge, we show that thanks to the central extensions of the Carroll algebra in the plane, and by considering systems with angular momentum, there exists a well defined notion of planar Carrollian statistical physics. Using Souriau's geometric thermodynamics, we compute the partition function for particles on a uniformly rotating disc, and show that rotation is inevitable for thermal equilibrium of planar Carroll systems, with one of the central charges determining the direction of rotation. We derive thermodynamic quantities in particular entropy, which scales logarithmically with the disc area, and pressure, which follows the two-dimensional ideal gas law. Though all results are obtained from symmetry considerations, we also derive the corresponding effective Hamiltonian.

We show that the Scarf I potential problem in a position-dependent mass background of the type $m(\alpha;x) = (1 + \alpha \sin x)^{-2}$, $0<\alpha<1$, can be solved by using a point canonical transformation mapping the corresponding Schr\" odinger equation onto that of the Scarf I potential with constant mass. The inverse point canonical transformation then provides some exactly-solvable rational extensions of the Scarf I potential with positive-dependent mass associated with $X_m$-Jacobi exceptional orthogonal polynomials of type I, II, or III. The Scarf I potential problem with position-dependent mass is shown to exhibit a deformed shape invariance property in a deformed supersymmetric framework. Such a property is also valid for extended potentials of type I and II. The results are illustrated with a simple example.

We study the mosaic Almost Mathieu operator, a quasiperiodic model that naturally admits a singular strip-Jacobi representation. By establishing a duality framework and extending the correspondence between the integrated density of states and the fibered rotation number to this setting, we obtain an effective reduction to $SL(2,\mathbb{R})$ cocycles. As a consequence, combining Aubry duality, reducibility theory, and the Moser--Pöschel argument, we prove that the spectrum is a Cantor set for all noncritical parameters.

We introduce an analytically invertible framework for wavefront construction based on the scattering properties of periodic waveguide networks governed by a gauge-shifted Helmholtz operator. By determining the exact transmission coefficients of the network, we express the lattice reactance as a Fourier expansion whose coefficients are analytically mapped onto the underlying graph architecture, allowing the required bond connections, refractive indices, lengths, and gauge phases to be determined directly from a prescribed target transmission coefficient. In contrast to conventional inverse-design approaches, the present formulation provides a closed-form route from desired wave transmission profiles to physically realisable structures. The framework extends naturally from one-dimensional angular filtering to two-dimensional image synthesis, where arbitrary transmitted intensity patterns are reconstructed through exact spectral control of the network scattering response.

In this paper, we study half-space observables in the massive Euclidean $\phi^4_4$ theory. We prove that the renormalized correlators can be multiplied by half-space truncations without requiring any additional renormalization. More precisely, products with smooth approximations of the half-space indicator converge to well-defined distributional limits, uniformly in the ultraviolet cutoff. The proof relies on a position-space renormalization framework for the Wilson--Polchinski flow equation based on a hierarchy of power-counting spaces adapted to the singularity structure of the correlators. This yields uniform power-counting bounds and Besov--Hölder regularity estimates for the renormalized correlators. As a consequence, the correlators converge as the ultraviolet cutoff tends to infinity, and the convergence takes place in explicitly identified Besov spaces.

We consider the class of Hopfield models of associative memory with activation function $F$ and state space $\{-1,1\}^N$, where each vertex of the cube describes a configuration of $N$ binary neurons. $M$ randomly chosen configurations, called patterns, are stored using an energy function designed to make them local minima. If they are, which is known to depend on how $M$ scales with $N$, then they can be retrieved using a dynamics that decreases the energy. However, storing the patterns in the energy function also creates unintended local minima, and thus false memories. Although this has been known since the earliest work on the subject, it has only been supported by numerical simulations and non-rigorous calculations, except in elementary cases. Our results are twofold. For a generic function $F$, we explicitly construct a set of configurations, called mixed memories, whose properties are intended to characterise the local minima of the energy function. For three prominent models, namely the classical, the dense and the modern Hopfield models, obtained for quadratic, polynomial and exponential functions $F$ respectively, we give conditions on the growth rate of $M$ which guarantee that, as $N$ diverges, mixed memories are fixed points of the retrieval dynamics and thus exact minima of the energy. We conjecture that in this regime, all local minima are mixed memories.

Global symmetry anomalies of a quantum field theory (QFT) can be packaged as specific couplings of a higher-dimensional symmetry theory (SymTh). In this work we show that for 5D superconformal field theories (SCFTs) engineered from M-theory backgrounds $X$ a Calabi-Yau cone, this data can be extracted from the path algebra of branes probing $X$. This provides a complementary algebraic approach compared with more geometric computations based on the explicit calculation of triple intersection numbers in a resolved geometry and / or $\eta$-invariants extracted from the boundary geometry $\partial X$. Our method applies in situations where the counterpart geometric computation is either unknown or combinatorially unwieldy. We illustrate with several toric threefold examples, including orbifolds $\mathbb{C}^{3} / \Gamma$ and more general non-orbifold Calabi-Yau cones of Sasaki-Einstein five-manifolds.

The Stanford-Witten-Norbury generalization of Mirzakhani's volume recursion computes $V^{(2m)}_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of $N=1$ supersymmetric Riemann surfaces of genus $g$ with $n$ Neveu-Schwarz boundaries of geodesic lengths $b_i$ ($i{=}1,\ldots,n$), and $2m$ Ramond punctures. Recently, a spectral curve has been derived that allows their Laplace transforms $W^{(2m)}_{g,n}(\{{\hat z}_i\})$ to be computed using topological recursion. We prove that the Stanford-Witten-Norbury volume recursion is directly derivable from the spectral curve. An alternative volume recursion can also be derived from it. The difference comes from whether the Ramond information is in the initial data, or in the volume recursion's kernels. The latter invites a geometrical understanding.

Efficient simulation of strongly-interacting fermionic systems on quantum processing units (QPUs) is a challenging task due to nonlocal mode entanglement generation. However, it is not yet well understood how the structure of entanglement governs the hardness of large-scale quantum chemistry simulations or the scaling of distributing such workloads. Here, we introduce an algorithm for estimating the distribution complexity of hybrid quantum-classical simulation for electronic structure Hamiltonians over heterogeneous high-performance architectures. Our algorithm relies on efficient analytical evaluation of the low entanglement boundaries for the orbital rotations and dephasing-induced localization within tensor fragments, in a double-factorized representation. Our entanglement estimation scales as $O(N^3)$ for each fragment, where $N$ is the number of orbitals. When QPUs are communicating via a quantum network, the cost of distribution per fragment is reduced quadratically from $O(N^2)$ to $O(N)$. Similarly, for hybrid quantum-classical approaches, with access to only conventional HPC interconnects, the worst-case cost is reduced from $O(\exp(N^2))$ to $O(\exp(N))$. We show that emergent entanglement patterns are induced by the interplay between coherent Gaussian orbital rotations and disordered Coulomb interactions. We discuss the underlying physical mechanisms that govern distribution complexity and introduce model systems that are tunable based on the localizability of fragments and the overlap of interfragment rotations. We characterize three different regimes of hardness for distribution complexity and classical simulability. The framework introduced here enables novel and more efficient quantum-classical application workflows towards utility-scale quantum computing.

We study a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions, and the solitons that emerge as the contrast $\delta$ tends to zero. Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of $\sqrt{\delta}$. Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in $\sqrt{\delta}$), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix. We illustrate the theory numerically and characterise a symmetry-breaking bifurcation in a symmetric dimer.

Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class $\mathbf{A}$ for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.

For negative values of the spreading coefficient (that is, in the so-called ``partial wetting'' regime), we prove that the thin-film equation with zero slip and repulsive potentials $P$ of the form $P(h)\approx h^{1-m}$ as $h\to 0$, $m>1$, admits for any positive speed a one-parameter family of travelling-wave solutions with a contact line and (as in standard slippage models) a logarithmically-corrected linear behaviour as $h\to +\infty$. These waves have locally finite rate of dissipation for any $m>1$ and locally finite energy for any $m\in (1,3)$. The result thus confirms that mildly repulsive potentials effectively resolve the no-slip paradox. The family is parametrized by a thermodynamically consistent contact-line condition which reduces to the classical fixed microscopic contact-angle one if $P\equiv 0$.

We investigate the dissipative dynamics of multilevel quantum systems strongly coupled either to a lossy cavity mode or directly to a bosonic environment. By deriving spectral localization bounds, we establish conditions under which strong system-bath coupling gives rise to a hierarchy of population relaxation timescales. Our approach builds on the reaction-coordinate polaron-transform framework. By mapping the original strong-coupling problem onto an effective weakly dissipative model, we analyze the spectrum of the resulting Liouvillian superoperator through localization bounds. For the generalized V model, we find that strong system-bath coupling gives rise to a bright-dark structure in the effective system-bath coupling operator: a single collective mode remains strongly coupled to the environment, while the remaining modes become progressively dark. Consequently, the dynamics separate into fast and slow sectors and, at finite coupling strengths, develop a hierarchy of population relaxation timescales. Numerical simulations based on both secular and non-secular quantum master equations corroborate the emergence of timescale separation and the pronounced slowing down of dissipative dynamics at strong coupling. Our results reveal a general mechanism underlying anomalously slow relaxation in strongly coupled open quantum systems and provide a route for engineering long-lived states through system-environment interactions.

Why do unrelated approaches to quantum gravity nearly all find spacetime two-dimensional at the shortest scales? Each theory answers only within its own dynamics; we highlight a single kinematic route to the same value, one assuming no field equation and living in the geometry of the space of states alone. That route is concentration of measure on the Fubini-Study geometry of pure states, which forces the pairwise distances of a random sample to equalize as the dimension grows, so any finite sample collapses to an equidistant dust whose thresholded metric graph is the complete graph. Handed this dust, a diffusion probe reads it as two-dimensional in the large-sample limit, the value the running spectral dimension takes at the dust's single relaxation scale, a property of the measurement rather than the structure; this convergence on two is not, by itself, evidence that spacetime is two-dimensional. Whether a given two is such an artifact is governed by the Laplacian spectrum near zero, and whether that reading carries across an emergence map is the condition we call spectral faithfulness; a single relaxation scale encodes no spectral dimension that tells one structure from another. The collapse, the probe value, and the eigenvalue-density criterion are machine-checked in Lean 4 against Mathlib, resting on the standard Beta law of overlaps; a power-law tail of small eigenvalues reads a genuine dimension, a single scale above a gap reads two at its own clock, and a gapped two-scale band reads off the universal line. These classes are run on graph-Laplacian proxies, and whether a link-graph reading carries to the physical nonlocal operator is left open. The spectral test reads the eigenvalue density near zero and separates, on a given structure, a measurement artifact from a dimension the structure genuinely expresses.

We study a piecewise constant function $\eta:\mathbb R\to\{-1,1\}$ with a finite number of discontinuities in any interval. We assume that the associated walk $\xi:\mathbb R\to\mathbb R$ satisfying $\xi'(x)=\eta(x)$, pinned by $\xi(0)=0$, has finite length excursions over past minima. This is the continuous generalization of an initial ball configuration in the discrete Box Ball System introduced by Takahashi and Satsuma, where solitons of integer sizes $k\ge1$ are identified. We extend the slot decomposition developed by Ferrari, Nguyen, Rolla and Wang in the discrete setting to the continuous case. Each soliton of $\xi$ is represented by a point in two dimensional space, one coordinate for position and the other for the soliton height, mapping $\xi$ to a point configuration. We consider a distribution on walks given by a product measure on the decomposition of the path into excursions over past minima. Excursions are distributed as products of their solitons weights, which are determined by the soliton heights. We show that when the weight function is in $L^1$ the slot decomposition of $\xi$ is a Poisson process. This extends to the continuous case an approach of Ferrari and Gabrielli. As an example, we compute the intensity measure of the Poisson process associated to the asymmetric telegraph process introduced by Kac. In a forthcoming paper we discuss the dynamic properties.

This paper presents a brief overview of studies of magnetic filament formation in the solar convection zone. Two stages of magnetic filament formation and development are distinguished. The first stage can be described within the kinematic approximation for the MHD equation, since the average kinetic energy of convective motion exceeds the average magnetic field energy. Moreover, magnetic Reynolds number is on the order of $10^6$. Therefore, at the initial stage of magnetic filament formation, the magnetic field can be considered frozen-in. It turns out that at this stage, magnetic filaments begin to form at the boundaries of convective cells in hyperbolic regions of the flow. These regions act as attractors for the magnetic field. At the free boundary of the convective zone near the interfaces between convective cells, the magnetic field has a predominantly normal component relative to the boundary. As the magnetic field increases, the field's frozen-in nature in the filament is disrupted, and saturation occurs due to finite conductivity. It is important to note that the magnetic field growth and saturation in the filament at this stage can be determined by analyzing the behavior of the magnetic field at the free boundary, which, according to observations, can be considered flat. Moreover, the formation of magnetic filaments is independent of the internal structure of the convective cells and is determined by the behavior of the convective flow along the free surface. In the next stage, when the magnetic field in the filament is sufficiently strong, the magnetic field pressure gradient begins to impede the convective flow, leading to a shift of the hyperbolic regions toward the convective flow along the free surface. As a result, the transverse size of the filament increases and stops when the kinetic energy density and the magnetic field energy density become comparable.

We extend the geometric electromagnetic particle-in-cell (PIC) framework, GEMPICX, to solve the quasineutral hybrid Vlasov-Maxwell equations with drift-kinetic electrons and fully kinetic ions. A structure-preserving finite difference method that employs dual grids is used. The discrete action principle for the hybrid model is derived, using the dual nature of the grids. The dynamical system for this hybrid quasineutral model does not explicitly involve the temporal evolution term for the electric field. A curl-curl equation is therefore used to implicitly obtain the component of the electric field that is parallel to the background magnetic field, at every timestep. The perpendicular component of the electric field is obtained using the quasineutral Ampere's equation without the displacement current, combined with the definition of the current in the drift-kinetic model. The discretized versions of the electric field equations are large, sparse linear systems. A fully explicit time-stepping scheme as well as two implicit-explicit (IMEX) schemes are tested. The numerical model is validated by verifying the various waves obtained from the dispersion relation.

We consider the classification of toric self-dual Einstein gravitational instantons with negative cosmological constant. As is well known, any Killing vector field on a self-dual Einstein manifold defines a local conformal Kähler structure. We prove that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions constructed by Calderbank, Pedersen and Singer.

We establish the calculus structure on Hochschild invariants of open-closed homotopy algebras. We further define the Getzler-Gauss-Manin connection and show that it is flat up to chain homotopy on the open-closed periodic cyclic chain complex.

We revisit the topic of false vacuum decay in field theory. We focus on a toy model of a real massive scalar field with an unstable quartic potential. This model has a false vacuum, and decay out of the false vacuum can be described via the method of constrained instantons, which work by introducing a constraint on the path integral. We identify and develop three different asymptotic limits which enable analytic construction of approximate {constrained} solutions. The first, in which the constrained solution is small compared to the inverse mass of the scalar field, is an application of the perturbative methods of Affleck, although we re-derive the main results and identify several terms which were previously neglected. Second, for very large constrained solutions we adapt the thin-wall approximation of Coleman. However, we find that the large instanton limit does not always exist. In this case we identify another useful limit, in which the Lagrange multiplier used to implement the constraint is large. In this limit, the solution's scaling with the parameters may be found via dimensional analysis and an exact solution is obtained with a single numerical computation.

A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

This paper pertains to the general theory of boundary value problems for general linear differential equations with smooth coefficients in a bounded domain with a smooth boundary and contains new advances in the general theory related to the boundary properties of solutions. Specifically, conditions on the traces of a solution to a general differential equation on the boundary of the domain are found and studied, allowing the solution to be reconstructed from its traces and the right-hand side of the equation. For the case of a general equation with constant coefficients, the resulting conditions on the traces of the solution take the form of a generalized moment problem.

Stationary wave functions at the integer quantum Hall transition are known to be multifractal, but the exact form of the multifractality spectrum has remained a subject of debate. While conformal field theory arguments predict a parabolic law, numerical simulations show deviations from parabolicity. We resolve this discrepancy by pointing out that powers of the local wave intensity fail to obey the Gaussian Free Field and Abelian Fusion Hypothesis assumed in earlier analysis. Rather, due to the non-Abelian nature of the underlying effective field theory, wave intensity correlators are dressed by insertions of background charge distributed uniformly in space. An exact expression for the $q$-moments of point-contact generated eigenstates is presented. Numerical tests are performed for the critical Chalker-Coddington network model on a rectangular torus. Our results are in precise agreement with the predictions of conformal symmetry realized as a level-4 current algebra.

The scientific utility of large single-dish radio telescopes depends critically on the stability and fidelity of their beam patterns, which govern angular resolution, sensitivity, and polarimetric accuracy. For the 32-m Ghana Radio Astronomy Observatory (GRAO) antenna, electromagnetic simulations reveal residual sidelobes, structural diffraction, and cross-polar leakage that limit performance in high-dynamic-range and polarisation-sensitive observations. To address these limitations, we develop a finite-impulse-response (FIR) spatial filtering framework that reformulates beam optimisation as a digital signal processing problem. By exploiting the equivalence between angular displacement and spatial frequency, classical FIR design methods, window-based and Parks-McClellan algorithms are adapted to operate directly on simulated Jones fields. This approach enables controlled suppression of high spatial frequency artefacts responsible for sidelobes and polarisation mixing, while preserving the telescope's diffraction-limited resolution. Applied to the GRAO 5 GHz beam model, the method achieves substantial reductions in near-in sidelobe ripple, improves beam smoothness, and lowers cross-polar leakage below -30 dB at boresight. These improvements translate into enhanced calibration stability and polarimetric precision, strengthening the telescope's capacity for Very Long Baseline Interferometry, spectral-line surveys, and pulsar timing. Beyond GRAO, the method provides a generalisable, non-invasive, and computationally efficient pathway for beam control applicable to other single-dish and phased-array instruments. The results establish digital spatial filtering as a practical complement to conventional optical or mechanical optimisation, advancing the integration of electromagnetic modelling and signal processing in next-generation radio astronomical instrumentation.

In this work, we study the fluctuations of a Mixture of Experts (MoE) generated by a quantum neural network trained via gradient flow on supervised learning problems. Our main results establish the Central Limit Theorem (CLT), and Sanov's principle for an MoE as the number of experts diverges. We demonstrate that the fluctuations of the empirical measure of its parameters close to its corresponding limit probability measure solve a linear transport equation. As a byproduct, we show that the MoE converges to a limit function which solves an evolution equation governed by the neural tangent kernel associated with the quantum neural network.

We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,\alpha}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.

We study the wave equation in the exterior of a strictly convex bounded domain $K \subset {\mathbb R}^d, d \geq 3,$ odd, with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $0 < \gamma(x) <1, \:\forall x \in \Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ In [10] we established that for $\gamma \equiv const$ and $K = \{x \in {\mathbb R}^3: \:|x| \leq 1\}$ the operator $G$ has no eigenvalues and we conjectured that the same result holds for every strictly convex obstacle. In this paper we prove this conjecture.

We investigate a discrete warped Chern-Simons description of three-dimensional warped gravity based on boundary holonomies and monodromy sectors. Starting from the lower-spin SL(2, R) + U(1) gauge structure associated with warped AdS(3) holography and warped conformal field theories (WCFTs), we construct a discrete boundary framework in which ordered products of link holonomies replace continuous gauge connections along noncontractible cycles. In this setting, boundary monodromies become the primary gauge-invariant observables characterizing the physical sectors of the theory. We show that the hyperbolic, eliliptic, and parabolic sectors naturally arise from the conjugacy classes of the discrete SL(2, R) monodromy, while the additional U(1) holonomy supplies the warped contribution to the boundary charges. Using these monodromy invariants, we derive a discrete entropy relation entirely from boundary holonomy data without relying on a smooth geometric thermal background. The resulting entropy reproduces the characteristic warped black-hole and WCFT structure in the continuum limit. We further demonstrate that the continuum warped holonomy conditions are recovered from the large-lattice limit of the ordered boundary products, establishing a direct correspondence between discrete monodromies and continuous Wilson loops. Our analysis suggests that warped gravitational thermodynamics may be understood from a fundamentally holonomy-based perspective in which boundary monodromy sectors provide an alternative organizational description of the physical states within the discrete warped framework. Keywords: warped Chern-Simons theory, boundary monodromies, holonomy sectors, Wilson loops, warped thermodynamics.

In this article, we establish geometric and analytic constraints imposed by the existence of nontrivial solutions to the Vafa-Witten equations on closed 4-manifolds. Using conformal invariance and refined Bochner-type estimates, we first prove an inequality relating the Yamabe constant $Y(g)$ to the $L^{2}$-norm of the self-dual Weyl tensor: $Y(g)\leq 2\sqrt{6}\|W_{g}^{+}\|_{L^2}$; when $Y(g)>0$, this yields a topological lower bound $\int_{M} |W_{g}^{+}|^{2} \geq \frac{4}{3}\pi^{2}(2\chi(M)+3\sigma(M))$. In the equality case, we show that the manifold must be Kähler with nonnegative scalar curvature and that the connection is reducible. As an application, for positive Einstein manifolds with $\operatorname{Ric}=3g$ admitting an irreducible Vafa-Witten solution, we obtain a sharp volume bound and prove the manifold cannot be Kähler. Through dimensional reduction $S^{1}\times N$, we establish a one-to-one correspondence between stable flat connections on a closed 3-manifold $N$ and $S^{1}$-invariant Vafa-Witten solutions, which yields a new estimate for the Yamabe constant $Y(g_{S^{1}\times N})\leq 2\sqrt{6\pi}\big(\int_{N}|\operatorname{Ric}(g_{N})-\frac{1}{3} R_{g_{N}}g_{N}|^2\big)^{1/2}$. Finally, under a regularity assumption that every anti-self-dual connection in the compactified moduli space is regular, we prove an energy gap: there exists $\varepsilon(g,P)>0$ such that any Vafa-Witten solution satisfies either $F_{A}^{+}\equiv0$ or $\|F_{A}^{+}\|_{L^{2}}\geq\varepsilon$.

We formulate an operator-algebraic mechanism by which exact Boolean records can arise from local completely positive quantum operations without being imposed as microscopic structure. The kinematic input is an algebraic process functional assigning probabilities to local normal completely positive operations in a finite operational context. From the predual response of a target algebra to source interventions, relative to a background strategy class, we define an influence algebra; exact events are then, by definition, the projections in its center. The dynamical question is whether nontrivial centers can be generated by coarse-graining rather than inserted through split-record laboratories. We address this question using state-preserving normal unital completely positive coarse-graining channels. If the Cesaro means of such a channel converge to a Choi-Effros infrared range and the range is asymptotically abelian in the GNS seminorm, then the represented infrared algebra is a commutative von Neumann algebra. Its projection lattice is therefore a complete Boolean algebra. We also give a finite-sector block-primitive criterion, motivated by locality and scrambling, which implies this asymptotic abelianness with exponential suppression of off-sector coherences and intra-sector fluctuations. The result is a conservative mathematical statement: classical facts are not identified with arbitrary projections of a Type-III local algebra, but with central projections selected by an asymptotically abelian completely positive infrared limit.

We investigate the transport and fluctuation properties of self-propelled particles whose motion is governed by an age-dependent phase-switching mechanism. The dynamics alternate between a Markovian downstream phase with a constant switching probability $r$ and a semi-Markovian upstream phase in which the age-dependent hazard probability $a/(b+c)$ decays with the internal clock $c$, representing persistent orientation. The time-averaged velocity, as an order parameter, shows a continuous transition at $a=1$ which separates an upstream-dominated ballistic regime ($a<1$) from an ergodic diffusive regime ($a>1$). Through generating-function methods and discrete-time moment recurrences, we derive exact expressions for the propagator and determine the long-time asymptotics of the mean displacement and variance. At the critical point $a=1$, the system exhibits giant fluctuations, with the variance scaling ballistically up to a logarithmic correction, $\mathrm{Var}(x_T) \propto T^2 / \log T$. These results demonstrate how slowly decaying reorientation probabilities lead to a marginal breakdown of the Central Limit Theorem, enabling unusually high-variance exploratory dynamics in biased environments.

A central goal of quantum optimal control is to achieve high-fidelity and low-energy control pulses. When quantum optimal control methods optimize every point of a pulse discretized over small time steps independently this can yield high fidelity control but also results in broadband and energy-hungry waveforms. We extend MAGICARP -- a shooting method inspired by Pontryagin's maximum principle on energy that generates an entire pulse from a small set of parameters, making it smooth and energy-efficient by construction -- from driftless systems to closed systems with the constant drift Hamiltonian of two exchange-coupled spins in an external magnetic field. The optimization proceeds in stages: the dressed states of the drift Hamiltonian structure the target, an initial shooting optimization is performed in the rotating-wave frame, and an exact laboratory-frame refinement follows. Benchmarked against Krotov and GRAPE at matched gate infidelity, MAGICARP consistently achieves the lowest energy and a conserved pulse area, concentrates its spectral weight on the gate-relevant transitions, and is the most robust to fluctuations in the exchange coupling; GRAPE independently converges to essentially the same pulse, while Krotov's method pays an order-of-magnitude energy premium. Moreover, a large statistical survey of unselected optimization runs resolves a weak-driving quantum speed limit for two exchange-coupled electron spins: low-amplitude realizations of the two-qubit quantum Fourier transform cease to exist below a critical gate time $T^*$ set by the drift's interaction rate, and the minimum control energy diverges on approach to this limit. The divergence obeys a simple two-parameter area-pole law, $E_2^{\mathrm{law}}(T)=A/T+B/(T-T^*)$, whose first term is the time-optimal area cost and whose second term is a pole at the speed limit.

We establish interior regularity results for a broad class of two-dimensional nonlinear elliptic systems. Our approach isolates the core integrability mechanism within a unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate. Specifically, we show that within any class of admissible pairs $(\boldsymbol{u}, \boldsymbol{f})$ that is stable under rescaling and satisfies a discrete oscillation-decay axiom, the map $\boldsymbol{u}$ is automatically locally Hölder continuous. Furthermore, the resulting Hölder exponent is explicit and optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term $\boldsymbol{f}$. This purely analytic framework systematically avoids the $\mathcal{H}^1$--$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization techniques that underpin existing regularity theories. We apply this principle to derive regularity results in regimes lying strictly beyond the reach of existing gauge-theoretic methods. As a foundational example, we provide a new direct proof of local Hölder continuity for almost harmonic maps $-\Delta \boldsymbol{u} = |\nabla \boldsymbol{u}|^2 \boldsymbol{u} + \boldsymbol{f}$ into $\mathbb{S}^n$ with $L^q$-integrable tension fields. We then extend the analysis to systems of the form $-\Delta \boldsymbol{u} = \Omega \cdot \nabla \boldsymbol{u} + \boldsymbol{f}$, replacing geometric antisymmetry assumption on the connection form $\Omega \in L^2$ with the purely analytic condition $\mathrm{div} \Omega \in L^q$ for some $q>1$...

State-dependent proto-area data produced by approximate recovery need not be compatible with a single local bulk metric. Using recovery maps calibrated on the code channel and held fixed along a logical-state family, we derive exact finite-resolution criteria and, near the hyperbolic disk, necessary and sufficient conditions for a regular proto-area two-jet to arise from a metric two-jet on a time-reflection-symmetric asymptotically AdS$_3$ slice. Finite networks give a polyhedral realization problem with primal and dual certificates, stable reconstruction, and explicit witnesses of nongeometry. In the continuum, the geometric tangent space is the range of the rank-two geodesic X-ray transform. A metric-forced Jacobi equation determines the normal Hessian of the renormalized boundary-length image and yields a gauge-invariant quadratic obstruction. Under a split-regularity hypothesis, nearby geometric data form a local graph; the two-jet criterion itself is unconditional for regular data. Hamiltonian-skewed codes realize both first-order nongeometry and a response whose first obstruction appears only at quadratic order. The compatible metric perturbation is reconstructed modulo boundary-fixing diffeomorphisms.

We give the complete solution of the one-parameter deformation of the six-vertex model on dynamical lattice introduced in [1] and dubbed gravitational seven-vertex model. The statistical model in question is mapped to a gas of self- and mutually avoiding loops on dynamical triangulations, with a temperature coupling controlling the volume not occupied by loops. The phase diagram is characterised by massive, dilute and dense critical phases, similarly to the gravitational O(n) loop model. There is however an important difference -- in our model the weights of the loops are not topological but depend on the form of the loop and on the curvature defects of the lattice via lattice spin connection. The seven-vertex model on dynamical lattice is nevertheless exactly solvable after being reformulated as a large-N matrix model, which we will refer to as 7vMM, and the solution in the scaling limit was found in [1]. Here we derive the full solution in terms of Jacobi theta functions and present the (non-algebraic) spectral curve of 7vMM in a parametric form. We obtain the phase diagram in the space of the two coupling constants -- the cosmological constant and the temperature -- and identify the critical phases along the boundary of the physical domain. We derive the scaling solution of [1] as the asymptotic of the full solution in the vicinity of the tricritical point separating the phases of dense and massive loops.

The vorticity distribution associated to canonical Gibbs point vortex ensembles under mean field scaling has Gaussian fluctuations for small negative inverse temperature. The perturbative argument is based on a cluster expansion of the partition function.

We show that an oscillon can be understood as a localized discrete resonant (non-normalizable) mode. Specifically, oscillon in the vacuum arises from the threshold mode, which because of nonlinearity gets localized. Following this idea, we find {\it wobblerons} - nonlinear excitations of kinks, that is, oscillons-kink bound state. Now, the oscillon can also originate in an antibound mode, i.e., a discrete, positive energy but non-normalizable mode.

Standard perturbative quantum field theory is persistently challenged by ultraviolet (UV) divergences and overlapping sub-divergences. In this work, we construct an absolutely UV-finite effective field theory (EFT) natively in four-dimensional spacetime. We embed the complete dynamical operator within a double-exponential capacity measure derived from quantized irreversible null-geometry. By analytically expanding the exact operator resolvent, we demonstrate that a universal macroscopic proper-time bound ($T \ge \tau_0$) natively emerges, structurally collapsing the high-order UV phase space via topological contraction. This mathematically defines the absolute UV boundary of the EFT. Consequently, overlapping subgraph divergences are rigorously identified as distributional pseudo-singularities caused by the continuous approximation analytically oversmoothing the discrete lattice bounds. These artifacts are systematically corrected by employing the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) protocol strictly as an algebraic mapping operator. Evaluating Quantum Electrodynamics in 4D, we obtain a finite electron bare mass ($m_0 \approx 0.427$ MeV) and derive finite renormalization constants, analytically preserving the exact Ward-Takahashi identity ($Z_1=Z_2$). In Quantum Chromodynamics, the exact preservation of Slavnov-Taylor identities yields a calculable finite bare strong coupling ($\alpha_{s,0} \approx 0.0191$). Generating analytically closed non-perturbative Schwinger-Dyson Equations, this framework provides a consistent, anomaly-free mathematical foundation for finite effective field theories to address dynamic chiral symmetry breaking and mass generation.

Witten's conformal boundary condition \cite{Witten:2018lgb} provides an elliptic boundary-value problem for the finite-boundary perturbative Euclidean gravitational path integral: one fixes the boundary conformal class and the mean curvature, while the trace-free extrinsic curvature is left free as the conjugate response. We show that this perturbative construction admits no half-supersymmetric extension in linearized minimal supergravity. For fixed conformal bosonic data, no half-dimensional gravitino boundary condition (local or pseudodifferential, APS-type included, with any compatible ghost condition at highest-derivative order) closes the full preserved chiral supersymmetry. Supersymmetry first selects the natural local chiral gravitino datum. Acting back on this datum then produces the trace-free extrinsic curvature, precisely the response that the conformal prescription leaves unfixed. The obstruction is therefore not the failure of a particular elliptic ansatz: even the chiral/Robin completion that is LS-elliptic and BRST-compatible at highest-derivative order would impose Dirichlet control on a Neumann response. The obstruction is pointwise in tangential momentum and survives compensating gauge transformations. It is a linearized, highest-derivative obstruction, not a global or nonlinear no-go; nonlinear supercovariant boundary terms may evade it by tying the trace-free extrinsic curvature to gravitino bilinears.

In this paper, using the example of a two-dimensional nonlinear sigma model with the Heisenberg group, we compare two variants of chiral field decomposition into a background part and a fluctuation. It is shown that only one of these methods is consistent with the construction of the generating functional by introducing a background field. Furthermore, we perform a one-loop renormalization of the quantum action, calculate power-law singularities in the two-loop approximation, and consider transition to an extended classical action. Finally, we study the consistency of the cutoff with special functional relations within the framework of the background field method.

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.

We derive a local bihamiltonian structure for the rational reduction of the 2D-Toda hierarchy (RR2T) of $(n,1)$-type by direct computations, and construct an $(n+1)$-dimensional semisimple generalized Frobenius manifold with non-flat unity whose Principal Hierarchy contains its dispersionless flows.

We develop an operator-frame formulation of entropic uncertainty relations in the Hilbert-Schmidt space of operators. For general continuous indexed operator frames, we derive an entropic uncertainty relation for the associated coefficient distributions by combining endpoint norm estimates with Riesz-Thorin interpolation. We then identify a distinguished class of mutually unbiased operator frames, defined through constant-modulus trace overlaps. Under suitable structural conditions, the corresponding coefficient amplitudes are related by a bilinear Fourier transform, leading to a stronger Hirschman-Beckner-type entropic uncertainty relation. As canonical realizations, we consider Weyl displacement operators and Wigner kernels, as well as Cartesian dyadic frames generated by position and momentum eigenstates. These examples recover familiar continuous-variable Fourier dualities while extending entropic uncertainty relations beyond measurement outcomes to operator representations themselves.

In this paper, we consider two classes of hyperkähler manifolds: moduli spaces of central-Levi parabolic Higgs bundles on the punctured sphere and star-shaped Nakajima quiver varieties. We produce a map $\mathcal T$ from a given star-shaped quiver variety $\mathcal X$ to a central-Levi parabolic Higgs bundle moduli space $\mathcal M$. We verify that $\mathcal T$ preserves stability and we show that it is a homeomorphism onto the locus of Higgs bundles with trivial underlying holomorphic structure. We then prove our main theorem: that $\mathcal T$ identifies the natural holomorphic symplectic structures on the two spaces. This theorem generalizes work by Biswas, Florentino, Godinho, Mandini from the rank 2, full flag, strongly parabolic case to arbitrary rank, partial flag, and weakly parabolic cases -- namely, those whose Higgs field residues project to the centers of their respective Levi subalgebras.

A random walk with small positive drift $\mu$ sets new records at a rate $\lambda(\mu)$ that vanishes as $\mu \to 0$. For centered steps attracted to a stable law $Y$ with index $1 < \alpha \leq 2$ and positivity parameter $\rho = P(Y>0)$, we find $\lambda(\mu) \sim K\mu^{(1-\rho)/\nu}$, $\nu=1-1/\alpha$, as $\mu \to 0$. The result is exact for Gaussian and strictly stable steps, and extends at the leading-power level to their domains of attraction. The exponent is set by the asymmetry only through $\rho$, sweeping the interval $[1,\,1/(\alpha-1)]$ as the skewness varies. It recovers the Gaussian linear law with slope $\sqrt{2}$ and, for symmetric heavy tails, the power $\mu^{\alpha/2(\alpha-1)}$; beyond the stable tail ratio, distributional details enter through the prefactor $K$, which is explicit for strictly stable steps. The result follows directly from one Mellin transform of the harmonic sum in the Spitzer-Baxter identity, which factorizes into a kernel transform and a Riemann $\zeta$ factor whose poles deliver at once the leading law, its prefactor, and a correction ladder, unifying diffusive, heavy-tailed, and skewed walks. The same transform also yields the expected maximum, recovering Kingman's heavy-traffic law for queues and Siegmund's corrected-diffusion constant as adjacent poles.

We consider expectations of the type $E\ \exp \left\{\sum_{i=1}^m \phi_i \right\}$, where $\phi_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ are functions, each depending on a few coordinates of a point in ${\Bbb R}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures. We prove sufficient conditions, in terms of the Lipschitz constants of $\phi_i$ and the combinatorics of their dependencies, for the integral to be separated from 0, and, consequently, to be amenable to a computationally efficient approximation. We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\Bbb R}^n$.

Quantum tunneling is expected to provide a computational speedup in quantum computing, a phenomenon that Adiabatic Quantum Optimization (AQO) aims to leverage. While some academic proofs of concept have been studied, such as the "Hamming weight with a spike" (HWS) problem, the algorithmic gains of this effect remain underexplored. In this work we extend the analysis underlying HWS to more general potentials. In the first half of the work, we establish (discrete) log-concavity of the ground state as a key structural property in this context. We devise a framework for establishing log-concavity of the ground state for a large family of discrete, 1-dimensional Schrödinger operators. The family includes convex potentials, but also certain potentials with local minima. In the convex case, this provides a discrete version of a continuous result by Brascamp and Lieb ('76). We demonstrate the utility of our result by establishing new spectral gap bounds, going beyond related results by Jarret and Jordan ('14) for convex potentials. In the second half of the work, we use our results on log-concavity to extend the perturbative analysis of HWS by Reichardt ('04) to the larger family of potentials with log-concave ground state. As a concrete instantiation, we use our result to extend the HWS analysis from a linear potential (which is exactly solvable) to a quadratic potential (which is no longer solvable). Our result strongly suggests the broader applicability of tunneling to convex potentials with spikes

Quantum instruments constitute the general description of quantum dynamics, encompassing both quantum measurements and quantum channels as special cases. Consequently, the incompatibility of quantum instruments represents a fundamental manifestation of nonclassicality in quantum theory. Here, we establish the operational significance of this notion by demonstrating communication tasks with classical inputs and outputs that enable the semi-device-independent certification of incompatible quantum instruments. We introduce a class of three-party communication tasks involving a sender, a relayer, and a receiver, and derive the tight upper bound of the figure of merits of these tasks achievable by all compatible instruments implemented by the relayer and this bound coincides with the optimal performance attainable in a classical communication subject to the same dimensional constraints. Violation of this bound certifies the incompatibility of the pair of quantum instruments implemented by the relayer. This identifies certification of incompatible instruments as a manifestation of quantum advantage in communication. This certification protocol is genuine as it is able to certify the incompatibility of a pair of instruments where the measurements and channels induced by the instruments are pairwise compatible and, therefore it does not depend on the incompability of measurements and channels induced by the instruments. Finally, we identify the simplest instances of our communication scenario that enable the certification of incompatible quantum instruments.

The quantum Zeno effect is a fundamental mechanism for implementing the effective dynamics of projected Hamiltonian and Lindbladian systems. It approximates the target projected evolution by interleaving Hamiltonian or Lindblad dynamics with quantum operations associated with the desired subspace. In contrast to the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is typically limited to order $1/n$. In this work, we improve this convergence rate by employing a multi-product formula, thereby achieving arbitrarily high-order convergence of the form $1/n^{K+1}$. This yields an improved approximation scheme for Zeno-like expectation values via an efficient post-processing method. The approach combines a modified Chernoff lemma, an adapted Dunford-Segal approximation, holomorphic functional calculus, and Chebyshev interpolation. We illustrate the method with the bosonic cat code and also consider the broader class of systems governed by the Bang-Bang decoupling method.

The May-Leonard model for three competing species, symmetric with respect to cyclic permutation of the variables and extended by diffusive terms, is considered. Exact time-periodic solutions of the system have been found, and their stability with respect to spatially periodic disturbances is studied. The stability of solu tions with respect to longwave spatial modulations is revealed. A period doubling instability breaking the spatial uniformity is found.

It is shown that any continuous function depending on several $p$-adic variables, each of which is defined on $\mathbb{Z}_{p}$, can be represented as a superposition of continuous functions of one $p$-adic variable. This statement is true for both functions with values in $\mathbb{R}$ and functions with values in $\mathbb{Q}_{p}$.

A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on spectral asymmetry can be encoded within and recovered from a negative order pseudodifferential operator -- the asymmetry operator -- constructed from appropriately defined pseudodifferential (spectral) projections. In this manuscript we apply these techniques to the study of the massless Dirac operator; in particular, we compute the principal symbol of the asymmetry operator, accounting for the underlying gauge invariance.

We introduce a non-Hermitian $\beta$-ensemble and determine its spectral density in the limit of large $\beta$ and large matrix size $n$. The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of Mezzadri and Taylor (2025). The joint distribution of eigenvalues contains a Vandermonde determinant to the power $\beta$ and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit $\beta\gg1$, our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large-$n$ limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum. The numerical study of the local nearest-neighbour spacing distribution shows agreement between the tridiagonal ensemble and two-dimensional Poisson statistics (independently of $\beta$), whereas we observe a $\beta$-dependence for the previously introduced ensemble.

A key test for any quasi-local energy in general relativity is that it be nonnegative and satisfy a rigidity property; if it vanishes, the region enclosed is flat. We show that the Hawking energy, also known as the Hawking mass, satisfies these properties under the dominant energy condition when evaluated on its natural area-constrained critical surfaces within a spacelike hypersurface (initial data set). In the time-symmetric case, these critical surfaces coincide with area-constrained Willmore surfaces, and we obtain positivity and rigidity theorems for the Hawking energy on such surfaces, including charged and cosmological constant (hyperbolic and spherical) variants as well as higher-dimensional analogues. In the fully dynamical (non-time-symmetric) case, we establish the first nonnegativity and rigidity theorems for the Hawking energy in this general setting. These results confirm the Hawking energy's consistency with basic physical principles and address several longstanding ambiguities and criticisms.

We present two constructions of continuum and thermodynamic limits of Abelian polyhedral gauge theories in arbitrary spacetime dimension. The first construction relies on the existence of projective systems of heat kernel measures, while the second involves an infinite dimensional heat kernel measure defined using projective limit of Hilbert spaces. As a special case, we obtain a model of Abelian gauge theory on the infinite cubical lattice, which, in contrast to the standard one, is massless for arbitrary values of the coupling parameter.

In this paper, we rigorously prove the bulk-edge correspondence for finite two-dimensional ergodic disordered systems. Specifically, we focus on the short-range Hamiltonians with ergodic disordered on-site potentials. We first introduce the bulk and edge indices, which are both well-defined within the Aizenman-Molchanov mobility gap. On the one hand, the bulk index is the usual Hall conductance, which is a well-studied quantized topological number. On the other hand, the edge index, which characterizes the averaged angular momentum of edge modes in the mobility gap, is uniquely associated with finite systems. Our main result proves that as the sample size tends to infinity, the edge index converges to the bulk index almost surely. Our findings provide a rigorous foundation for the bulk-edge correspondence principle for finite disordered systems. The existence of the Aizenman-Molchanov mobility gap is proved by the geometric decoupling method, introduced by Aizenman and Molchanov [Comm. Math. Phys., 1993], under a rational assumption on the distribution of the random potential. For completeness, all assumptions are checked on a prototypical model for (quantum) anomalous Hall physics.

Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the İnönü--Wigner construction starts with a Lie algebra $\mathfrak{g}$ with Lie subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ and complement $\mathfrak{n}$. Then, the vectors in $\mathfrak{n}$ are rescaled by a formal parameter $\varepsilon \in \mathbb{R}_+$, which effectively turns the Lie bracket $[ \, \cdot \, , \cdot \, ]$ into an $\varepsilon$-dependent family $[ \, \cdot \, , \cdot \, ]_\varepsilon$. Notably, the limit $\varepsilon \to 0$ trivializes certain relations, such that the complement $\mathfrak{n}$ becomes an abelian ideal. In the present article, we are not only interested in the limiting Lie algebras and groups, but also in the corresponding power series expansions in $\varepsilon$ to understand their limiting behavior. Particularly, we are interested in the integration of the `power-series-expanded' Lie algebras to their corresponding Lie groups. To this end, we reformulate the above procedure using infinite-dimensional Lie algebras of analytic germs and then apply their integration theory. Our main results are a construction of the corresponding `Lie group expansions' in terms of quotients of groups of analytic germs and an explicit description of these groups in elementary terms. Applications of this procedure include the geometric Newtonian limit of General Relativity to Newton--Cartan gravity, where the Poincaré group is contracted to the Galilei group.

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.

Starting from the following multidimensional integrable generalizations of the heavy rigid body systems: the Euler top, the Lagrange top, the Lagrange bitop, and the totally symmetric case, we add to each of them a gyroscope. For each of the newly constructed systems, we provide a polynomial matrix Lax representation and prove Liouville integrability.

It is well known that the diffusion equation, when treated as a stand-alone partial differential equation, exhibits exponential instabilities in boosted frames, which render the corresponding initial-value problem ill-posed. Recently, however, it was shown that Fick-type diffusion arises as the exact hydrodynamic sector of relativistic Fokker-Planck kinetic theory. In this work, we exploit this kinetic embedding to formulate a modified initial-value problem for one-dimensional Lorentz-boosted diffusion. We show that the resulting dynamics are well posed both forward and backward in time, provided the boosted density profiles admit a kinetic-theory realization. Such profiles form a space of band-limited functions, within which the evolution can be expressed as a discrete superposition of spatially sampled initial data, weighted by a Shannon-Whittaker-type Green function defined on the full Minkowski plane. The Green function is obtained in closed analytic form.

We investigate numerically the blocking of two-dimensional bistable reaction diffusion fronts by geometric obstacles. Our goal is to derive quantitative criteria for front propagation in the presence of spatial heterogeneities. Using a conservation law approach, we show that the integral of the reaction term acts as an effective driving force for the front. Combining this insight with the exact one-dimensional traveling wave solution, we construct a reduced analytical model that predicts blocking thresholds. In particular, we obtain explicit conditions for front propagation in a waveguide connected to a conical region of angle theta, valid for widths w less than 4. The model captures the influence of both geometry and nonlinearity, and shows good agreement with numerical simulations. Finally, we extend the analysis to more complex geometries, including checkerboard-like obstacles, and derive simple heuristic rules governing front propagation. ~

We exhibit an exact correspondence between sampling with score-based diffusion models and adiabatic transport of ground states for a family of Schrödinger operators we call Score Hamiltonians, built from the learned score's quantum potential. We obtain novel density reconstruction bounds and principled annealing schedules via adiabatic theorems for Fokker-Planck equations with time-varying potentials. We find the fundamental limit of sampling is set by the ratio of squared score-matching error to Score Hamiltonian spectral gap - the inverse Poincaré constant of the data density.

Random non-hermitian matrix ensembles with double-sided rotation invariance obey, in the limit of large matrix size, the Single Ring Theorem, which states that the support of the mean eigenvalue distribution in the complex plane is either a disk or an annulus. In contrast, rotational-invariant random normal matrix ensembles can have mean eigenvalue densities supported over any number of concentric annuli in the complex plane. In this paper we introduce and investigate, both analytically and numerically, a non-hermitian matrix model which flows from a generic matrix distribution obeying the Single Ring Theorem to a distribution of normal matrices by tuning a parameter which penalizes non-normality. We observe numerically breakdown of the Single Ring Theorem as the model flows towards normality, and determine the critical value of the parameter at which the transition occurs. We also study in detail the behavior of the singular values of these matrices under the flow. These singular values form a Fermi gas confined to the positive half-line. In particular, we find that at small values of the flow parameter, the interparticle spacings in the gas exhibit Wigner-Dyson repulsion, whereas for asymptotically large values of the flow parameter, at the normal matrix endpoint of the flow, the spacing statistics is Poissonian. The flow interpolates continuously between these two types of statistics. However, this change in statistics is not related directly to breaking of the Single Ring Theorem, which occurs very early-on along the flow, in the regime of Wigner-Dyson statistics. Finally, we introduce a certain ensemble of random permutations associated with the gas, and make a conjecture on how to use it in order to reconstruct approximately the average density of complex eigenvalues from that of the singular values in the large-$N$ limit.

We prove a holomorphic residue localization formula for odd holomorphic vector fields on compact complex supermanifolds whose fermionic and bosonic dimensions coincide. Under isolated non-degeneracy hypotheses on the reduced zero set, we give an explicit local residue formula.

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that for a predominant metric, the number of closed geodesics of length smaller than $T$ has a stretched exponential upper bound in $T$. In addition, we study remainders in the Weyl law for predominant metrics. The Weyl law states that the number of Laplace-Beltrami eigenvalues smaller than $\lambda^2$ is asymptotic to $C\lambda^d$ with an $O(\lambda^{d-1})$ error. We show that, for a predominant metric, the estimate on the error can by improved by a power of $\log \lambda$. After an application of recent results of the authors in the case of the Weyl law, these estimates follow from a study of the non-degeneracy properties of nearly closed orbits for predominant sets of metrics.

We study conformal blocks of vertex operator algebras on configuration spaces from the viewpoint of the parenthesized braid operad, a combinatorial model of the fundamental groupoid of the little 2-disk operad. For each binary tree we introduce coordinates and a simply connected domain in the configuration space, and show that conformal blocks admit convergent expansions on these domains. Inserting one binary tree into a leaf of another gives gluing maps for the corresponding conformal blocks, while analytic continuation along paths in configuration spaces gives isomorphisms between conformal blocks associated with different trees. We prove that these operations are compatible with the operadic composition in the parenthesized braid operad. As a consequence, the category of $C_1$-cofinite modules whose contragredient modules are finitely generated carries a canonical unital pseudo-braided category structure, without assuming rationality or $C_2$-cofiniteness of the vertex operator algebra. In the rational $C_2$-cofinite case, this structure is represented by tensor products and recovers the balanced braided tensor category structure with twist $\exp(2\pi iL(0))$.

This paper examines the Maxwell system of electrodynamics within the framework of distributions. A primary objective is to establish boundary conditions for fields at interfaces when the charge and current densities are measures localized on the interface. From this analysis, the paper presents a derivation of the generalized Snell's law, along with formulas for the amplitudes of the reflected and transmitted waves in terms of the incident amplitude.

This article is devoted to the study of a finite system of long clusters of subcritical 2-dimensional FK-percolation with q $\geq$ 1, conditioned on mutual avoidance. We show that the diffusive scaling limit of such a system is given by a system of Brownian bridges conditioned not to intersect: the so-called Brownian watermelon. Moreover, we give an estimate of the probability that two sets of $r$ points at distance $n$ of each other are connected by distinct clusters. As a byproduct, we obtain the asymptotics of the probability of the occurrence of a large finite cluster in a supercritical random-cluster model.

We study existence, nonexistence, and qualitative properties of ground states for a focusing, subcritical Nonlinear Schrödinger Equation on a hybrid plane, consisting of a half-line attached to a plane. Ground states are normalized minimizers of the associated energy, given by Nonlinear Schrödinger energies with contact interactions on the half-line and on the plane, plus a quadratic coupling term. At fixed mass, existence holds if the contact interaction on the half-line is not too repulsive, or the interaction on the plane is sufficiently attractive, or the coupling is strong enough. Nonexistence occurs when both interactions are sufficiently repulsive and the coupling is weak. Moreover, we discuss how the coupling affects the support and the symmetry properties of such ground states. These are the first results for a Nonlinear Schrödinger Equation on a mixed-dimensional manifold.

We demonstrate that the complete and non-redundant set of Landau singularities of Feynman integrals may be explicitly obtained from the Whitney stratification of a certain map. As a proof of concept, we leverage recent theoretical and algorithmic advances in their computation in order to determine this set for nontrivial examples of two-loop integrals. Interestingly, different strata of the Whitney stratification describe not only the singularities of a given integral, but also those of integrals obtained from kinematic limits, e.g. by setting some of its masses or momenta to zero.

This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace-Beltrami operator is presented and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.

We pursue the current developments in random tensor theory by laying the foundations of a free probability theory for tensors and establish its relevance in the study of random tensors of high dimension. We give a definition of freeness associated to a collection of tensors of possibly different orders. Our definition reduces to the usual freeness when only tensors of order 2 are concerned. We define the free cumulants which are associated to this notion of tensor freeness. We prove that the basic models of random tensors are asymptotically free as the dimension goes to infinity. On the way, we establish Schwinger-Dyson loop equations associated to random tensors.

Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n \times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n \times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n \times 2n$ real symmetric matrices denoted by $\operatorname{EigSpSm}(2n)$. Our perturbation bounds on symplectic eigenvalues for $\operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}.

We establish the exponential clustering of correlation functions for the high-temperature Gibbs states of Bose-Hubbard type models. To overcome the technical difficulties arising from the unboundedness of bosonic operators, we develop the interaction-picture cluster-expansion technique. This method also allows us to systematically bound the moments of the local particle number. This result provides an analytical justification for the low-boson-density condition frequently assumed in the study of bosonic many-body systems. As direct mathematical consequences of the clustering property, we derive a uniform upper bound on the specific heat density and establish a bosonic thermal area law with improved temperature dependence.

The symmetric inclusion process (SIP) models particles diffusing on a graph with mutual attraction. We recently showed that, in the log-concave regime (where diffusivity dominates interaction), the spectral gap of the conservative SIP matches that of a single particle. In this paper, our main result demonstrates that this identity generally fails outside this regime, but always holds for the non-conservative SIP, regardless of the interaction strength. When this one-particle spectral gap identity breaks down, we derive sharp bounds for the gap in terms of diffusivity, and reveal a two-particle spectral gap identity in the vanishing diffusivity limit. Our approach leverages the rigid eigenstructure of SIP, refined comparisons of Dirichlet forms for arbitrary diffusivity and particle numbers, and techniques from slow-fast system analysis. These findings extend to the dual interacting diffusion known as Brownian energy process, and shed some light on the spectral gap behavior for related Dirichlet-reversible systems on general, non-mean-field, geometries.

In the study of distributed quantum information processing, it is a fundamental problem to optimize local operations in the implementation of non-local quantum operations assisted by limited entanglement. We develop an algebraic-geometric framework that systematically simplifies optimization over separable (SEP) channels -- widely used as approximations of local operations -- and strengthens the Doherty--Parrilo--Spedalieri (DPS) hierarchy for solving such problems. We apply this framework to computing maximum success probability for exactly implementing a broad range of different non-local operations under SEP channels. First, we present a unified generalization of previous analytical results on the entanglement cost. Via the generalization, we resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. Second, we numerically determine the trade-off between the strength of entanglement and the success probability of implementing various operations -- such as entanglement distillation, non-local unitary channels, measurements, and state verification.

We construct and study a bicategory of super 2-line bundles over graded Lie groupoids, providing a unified framework for geometric models of twistings of (real) K-theory. The core of our work is to exhibit a wide range of models from the literature as special cases, among them several variants of bundle gerbes (real/equivariant/Jandl), Freed-Moore's twisted groupoid extensions, Freed-Hopkins-Teleman's K-theory twistings, Moutuou's real twistings, Freed's invertible algebra bundles, and Distler-Freed-Moore's orientifold twistings.

We investigate the entanglement entropy (EE) and spectral gap properties of highly entangled spin chains arranged along a Hamiltonian path on a two-dimensional (2D) lattice with geometries reminiscent of Fermat's spiral. Interpreting the interactions along the spin chain as the strongly anisotropic limit of a 2D model, with couplings oriented along different directions in different quadrants, we construct an exactly solvable ground state (GS) that exhibits volume scaling of EE across bipartition through the center in any direction. This provides another mechanism for realizing 2D GSs with local interactions that violate the entanglement area law. As in the previously studied coupled-chains paradigm, the new construction features an entanglement phase transition, but with distinct scaling at the critical point and in the weakly entangled phase, and a faster closing of the spectral gap in the highly entangled phase. The corresponding tensor network representation uses lower-rank tensors while preserving a global geometry similar to that of coupled-chains model. Finally, the Fermat-spiral layout naturally generalizes to two highly entangled 1D chains coupled by a quantum junction at the center of the 2D system.

Tensor renormalization group, originally devised as a numerical technique, is emerging as a rigorous analytical framework for studying lattice models in statistical physics. Here we introduce a new renormalization map - the 2x1 map - which coarse-grains the lattice anisotropically by a factor of two in one direction followed by a 90-degree rotation. We develop a novel graphical language that translates the action of the 2x1 map into a system of inequalities on tensor components, with rigorous estimates in the Hilbert-Schmidt norm. We define a finite-dimensional "bounding box" called the hat-tensor, and a master function governing its RG flow. Iterating this function numerically, we establish convergence to the high-temperature fixed point for tensors lying within a quantifiable neighborhood. Our main theorem shows that tensors with deviations bounded by 0.02 in 63 orthogonal sectors flow to the fixed point. We also apply the method to specific models - the 2D Ising and XY models - obtaining explicit bounds on their high-temperature phase. This work brings the Tensor RG program closer towards a rigorous, computer-assisted construction of critical fixed points.

We present a construction of the double copy for gauge theories that exhibit off-shell color-kinematics duality within the Batalin-Vilkovitsky (BV) formalism. As illustrative examples, we consider the double copies of Chern-Simons theory, four-dimensional BF theory, and two-dimensional Yang-Mills theory, and we discuss possible gravity interpretations for these cases. We formalize the construction and demonstrate that Kodaira-Spencer gravity, Kähler gravity, and their generalizations, fit naturally within this framework. In particular, Kodaira-Spencer gravity emerges as a gauge theory describing deformations of generalized complex structures, while the double copy of Chern-Simons theory captures the deformation of the Courant bracket.

The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system's boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.

The cylindrical Hastings-Levitov$(0)$ admits a single infinite connected tree (arm). For a cylinder of width $N$ and particles of size $\lambda$, {we consider the first time $\upsilon_{N, \lambda}$ after which only the unique infinite tree receives particles}. We prove that $\frac{cN^2}{\lambda^3} \le \mathbb{E}[\upsilon_{N, \lambda}]\le\frac{CN^2}{\lambda^3}$, and establish an exponential tail for $\upsilon_{N, \lambda}$. Moreover, we obtain an asymptotic bound to the expected total number of trees, and the last time a new tree emerges.

Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for bipartite compiled Bell games via a newly formalized convergent sequential Navascués-Pironio-Acín (NPA) hierarchy: any polynomial-time prover's score is controlled by a finite-level hierarchy value, and finite-level convergence gives a negligible gap to the commuting quantum value, or to the tensor-product quantum value under flat optimality. Second, we provide a full characterization of this sequential NPA hierarchy, establishing it as a robust numerical tool that is of independent interest. Finally, for games without such finite-level certificates, we explore the necessity of NPA approximation error for quantitatively bounding their compiled scores, linking these considerations to the complexity conjecture $\mathrm{MIP}^{\mathrm{co}}=\mathrm{coRE}$ and open challenges such as quantum homomorphic encryption correctness for "weakly commuting" quantum registers.

We prove that for two-marginal optimal transport with Coulomb cost on $\mathbb{R}^d$, the optimal map is a $C^{1,\alpha}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $\alpha$-Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains).

In this paper we investigate analytically the formation of finite time singularities in the three dimensional incompressible Euler equations under the model of Gibbon, Fokas, and Doering for vorticity stretching within a bounded cylindrical domain and under axisymmetric conditions. We derive explicit Lagrangian solutions for the vorticity, its stretching rate, fluid pathlines, and velocity components by exploiting constants of motion associated with the field dependent infinitesimal symmetries of the system. The central finding is that the existence and nature of a finite time singularity are determined exclusively by the local geometric structure of the initial vortex stretching rate near its global minimum. Whether a singularity forms depends on how flat this profile is at the minimum. Flatter profiles delay the blowup and sufficient flatness can suppress it entirely. For power law behavior near the minimum, critical thresholds for the exponent are identified which separate regular solutions from those that develop a finite time singularity. These thresholds differ depending on whether the singularity occurs at the centre of the cylinder or on a ring away from the centre, with minima at the centre requiring higher flatness to avoid blowup. This work provides a rigorous analytical framework that elucidates how the local geometric structure of the initial conditions governs the potential for singularity formation in 3D fluid flows, offering fundamental insights into the interplay between symmetry, initial data, and the development of extreme events in idealised turbulence.

We construct symmetry topological field theories (SymTFTs) using the sandwich construction of Pulmann-Ševera-Valach that manifest the centre symmetries of Chern-Simons theory and Yang-Mills theory as well as general relativity in the MacDowell-Mansouri formulation. The 'filling' of the sandwich is an AKSZ sigma model whose target space is a Weil algebra, augmented with discrete degrees of freedom given by a choice of topological boundary condition.

Compiled nonlocal games transfer the power of Bell-type multi-prover tests into a single-device setting by replacing spatial separation with cryptography. Concretely, the KLVY compiler (STOC'23) maps any multi-prover game to an interactive single-prover protocol, using quantum homomorphic encryption. A crucial security property of such compilers is quantum soundness, which ensures that a dishonest quantum prover cannot exceed the original game's quantum value. For practical cryptographic implementations, this soundness must be quantitative, providing concrete bounds rather than merely asymptotic. While quantitative quantum soundness has been established for the KLVY compiler in the bipartite case, it has only been shown asymptotically for multipartite games. This is a significant gap, as multipartite nonlocality exhibits phenomena with no bipartite analogue, and the difficulty of enforcing space-like separation makes single-device compilation especially compelling. This work closes this gap by providing quantitative upper bounds for all multipartite compiled nonlocal games via a new sequential NPA-like hierarchy. In particular, finite-level convergence yields quantitative quantum soundness with respect to the commuting quantum value, and flat optimality yields the same with respect to the tensor-product quantum value. On the way, we introduce an NPA-like hierarchy for quantum instruments and prove its completeness, thereby characterizing correlations from operationally-non-signaling sequential strategies. This NPA-like hierarchy can be seen to complement previous multipartite generalizations of the S-G-HJW purification theorem, which takes a central role in quantum information, nonlocality, and contextuality. We further develop novel geometric arguments for the decomposition of sequential strategies into their signaling and non-signaling parts, which might be of independent interest.

Fermat's principle is fully generalized to the case where a smooth interface separates two cone structures -- Lorentz-Finsler lightcones -- representing wave propagation in a potentially inhomogeneous, anisotropic, time-dependent and discontinuous medium. The interface, wave source and receiver are assumed to be a hypersurface, a submanifold and a curve in the spacetime, respectively, of any causal character. For a trajectory to fullfil Fermat's principle -- i.e., to be a critical point of the arrival time functional -- its direction must change at the interface, obeying a precise condition that generalizes Snell's law of refraction when the wave crosses the interface, or the law of reflection when it remains in a single medium. Both laws are analyzed in detail to establish the conditions ensuring the existence and uniqueness of refracted and reflected trajectories, and to determine whether they actually minimize the arrival time. Applications to Zermelo's navigation problem and the determination of geodesics in discretized spacetimes are also emphasized.

Given a semi-simple algebra equipped with a coproduct, the Clebsch-Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the Clebsch-Gordan coefficients of the Lie algebra $\mathfrak{sl}_2$ are given in terms of the dual Hahn polynomials. Taking the reversed point of view, we show that any finite dimensional family of polynomials belonging to the Askey scheme can be interpreted as Clebsch-Gordan coefficients of an algebra. The Hahn polynomials are thus associated to the oscillator algebra with the Krawtchouk polynomials treated through a limit. The dual Hahn polynomials and Racah polynomials are seen to be associated to $\mathfrak{sl}_2$ with a more general coproduct than the standard one. The $q$-Hahn polynomials are interpreted as Clebsch-Gordan coefficients of a $q$-deformation of the oscillator algebra and the $q$-Racah polynomials are seen to be connected in this way to ${\rm U}_q(\mathfrak{sl}_2)$ with a generalized coproduct.

We investigate the stability of synchronized oscillations in coupled nonlinear systems by establishing a spectral correspondence between continuous linear shift-invariant (LSI) media and discrete networks. In this framework, Fourier modes of a continuous spatial operator and eigenmodes of a network coupling matrix are treated as spectral parameters of the same Master Stability Function. This correspondence allows finite-wavenumber instabilities of continuous media to be translated into predictable instability windows in network coupling space. Applying the framework to zero-row-sum Metzler coupling matrices and using a competitive Lotka-Volterra model as a paradigm, we show that synchronization may exhibit reentrant behavior: it is stable for weak coupling, lost within intermediate coupling intervals, and restored at stronger coupling. The framework also reveals a distinction between undirected and directed networks. For undirected networks, the relevant spectra are real and the resulting instability mechanism is analogous to that of standard reaction-diffusion systems with real wavenumbers. Directed networks, however, can possess complex spectra. We show that such complex spectral modes can induce quasiperiodic bifurcations of the synchronized state, leading to dynamical regimes that are inaccessible to standard real-wavenumber reflection-invariant reaction-diffusion models.

The recently introduced random purification channel, which converts $n$ i.i.d. copies of any mixed quantum state into a uniform convex combination of $n$ i.i.d. copies of its purifications, has proved to be an extremely useful tool in quantum learning theory. Here we give a remarkably simple construction of this channel, making its known properties -- and several new ones -- immediately transparent. In particular, we show that the channel also purifies non-i.i.d. states: it transforms any permutationally symmetric state into a uniform convex combination of permutationally symmetric purifications, each differing only by a tensor-product unitary acting on the purifying system. We then apply the channel to give a one-line proof of (a stronger version of) the recently established Uhlmann's theorem for quantum divergences.

We study deformations of a genus one Riemann surface and of a second order Abelian differential on the surface which preserve the periods of the differential with respect to a chosen canonical homology basis of the surface. We call these deformations isoperiodic. We derive a second order ordinary differential equation with rational coefficients governing the variations of the position of the unique pole of the differential under the isoperiodic deformations. The obtained equation depends on the order of the pole of the differential. We characterize the solutions of the obtained ordinary differential equations that correspond to the isoperiodic deformations. We apply these results to the theory of genus one solutions to the Boussinesq equation.

We investigate the topological entanglement entropy of quantum states arising in the context of three-dimensional Chern-Simons theory with compact gauge group $G$ and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{dm,dn}$ torus link complements, which is a $d$-party pure quantum state, and analyze its large-party limit, i.e., $d\to \infty$ limit. We show that the entanglement measures in this limit will receive contributions only from the Abelian anyons, and non-Abelian sectors are suppressed in the large-party limit. Consequently, the large-party limiting value of the entanglement entropy has an upper bound of $\ln |Z_G|$, where $|Z_G|$ is the order of the center of the group $G$. As an explicit example, we perform quantitative analysis for the simplest case of the SU(2) group and $T_{d,dn}$ torus link to obtain the large-party limit of the entanglement entropy. We further investigate the semiclassical ($k \to \infty$) limit of the entropies after taking the large-party limit for this particular example.

We study the abelian sub-C*-algebra of the CAR algebra generated by the start and face opertors of Kitaev's toric code. We show that it is a C*-diagonal equivalent to the canonical diagonal of the CAR algebra.

This paper is devoted to the quantum integrable structure of Wess-Zumino-Novikov-Witten models, formed by an infinite number of commuting Integrals of Motion (IMs) in their current algebra. Focusing for simplicity on the SU(2) case, we obtain the first four commuting higher-spin local IMs, starting from a general SU(2)-invariant ansatz and imposing their commutativity. We further show evidence of their commutativity with quantum non-local IMs, which were already built in the literature as Kondo defects. We then investigate the diagonalization of these local operators on $\widehat{\mathfrak{su}(2)}_k$ Verma modules: we explicitly find the first few eigenvectors and further discuss the affine Bethe ansatz and ODE/IQFT conjectures, which predict the full eigenstates and spectrum of the integrable structure. Our results show a perfect match between the direct diagonalization and these overarching conjectures. We conclude by discussing several outlooks, including multi-current generalisations, massive deformations and a general long-term program towards the first principle quantisation of 2-dimensional integrable sigma-models.

Distinguishability is central to quantum information theory, but a quantitative measure for distinguishing topological defects--realizations of generalized symmetries in quantum field theory (QFT)--has been lacking. We introduce the notion of the \textit{defect relative entropy} to fill this gap for topological defects in two-dimensional conformal field theories (CFTs). For defects on a circle, we derive a universal formula that reduces defect relative entropy to a Kullback--Leibler divergence determined entirely by the modular $\mathcal{S}$-matrix and defect coefficients. Thus, the algebraic data governing modular transformations also determines defect distinguishability. A striking consequence is that certain distinct defects can have vanishing relative entropy when restricted to one side, implying that an observer confined to that side cannot distinguish them. This gives rise to information-theoretic equivalence classes of defects, which we term \textit{defect relative sectors}. We further introduce the sandwiched defect Rényi relative entropy and defect fidelity, derive general formulas for these quantities. Explicit calculations in the Ising model, tricritical Ising model, and $\widehat{su}(2)_k$ WZW models illustrate our results.

Starting from the Navier--Stokes equations in rotating spherical coordinates with depth-varying density and eddy viscosity, we derive an asymptotic model describing non-equatorial wind-generated ocean drift currents. Our approach allows for large-scale flows that cannot be captured by classical tangent-plane approximations. The strategy is to perform a careful scaling and to perform a double asymptotic expansion with respect to two small parameters arising from the scaling: the Rossby number and the ratio between the Ekman depth and the Earth's radius. We obtain a system of linear ordinary differential equations with nonlinear boundary conditions governing the leading-order dynamics, highlighting that the dynamics is governed by the linear terms, whereas the nonlinear ones, related to the injection and dissipation of kinetic energy, appear only at higher order. We use the leading-order equations to compare our model with the simplest theory of ocean circulation due to Sverdrup and note that, even at this level of simplification, our equations have the potential to provide deeper insight. Subsequently, focusing on Ekman flows, we prove existence and uniqueness of the leading-order solution, which retains the classical Ekman spiral structure for arbitrary eddy viscosity profiles. Finally, we compute the surface deflection angle of the wind-driven current for three explicit eddy viscosity profiles, obtaining results consistent with observations. In addition, we derive the governing equations for the first-order correction with respect to the Rossby number and provide a priori bounds for its solution.

We investigate a dual variational formulation, in the spirit of Brenier, for several compressible fluid models: the compressible barotropic Euler system, the quantum Euler system, and the Euler-Korteweg system. We identify a unified abstract framework encompassing all three systems, which enables a simultaneous analysis. By introducing time-adaptive weights, we establish the consistency of the duality scheme on large time intervals. We prove the existence of variational dual solutions to the corresponding Cauchy problems for continuous, vacuum-free initial data in spaces of finite Radon measures, and establish the absence of a duality gap. As an application, we formulate and prove a 'Dafermos principle' for these models: no subsolution can dissipate the total entropy earlier or at a faster rate than the corresponding strong solution on its interval of existence. We also discuss connections between our abstract consistency result and Brenier's shock-free substitutes for entropy solutions of Burgers' equation.

We develop a categorical and algebro-geometric treatment of localization for cohomological theories endowed with an open--closed recollement. Starting from a class whose restriction to the open complement vanishes, we show that the natural output of the formalism is, in general, not a distinguished localized class on the closed locus, but a torsor of supported refinements. This torsor is the secondary, pre-denominator object of localization; categorically, it gives rise to a translation groupoid of supported lifts. A canonical local term appears only after an additional uniqueness or concentration principle is imposed. We construct the torsor from the localization triangle and establish excision, Cartesian base-change pullback, proper pushforward, and compatibility with external products under explicit hypotheses. We also prove that any assignment of local terms compatible with the localization triangle factors through this torsor. With Verdier duality and orientation data, supported refinements define local indices and global-to-local index formulas. Under purity and concentration, the torsor is rigidified and the familiar Euler-denominator expressions are recovered.

Causality -- that a response cannot precede its cause -- is among the most universal constraints in physics. Yet when a unitary microscopic theory is reduced to an open-system memory-kernel equation, causality is not inherited for free: an upper-half-plane pole of the kernel forces exponential growth of the reduced propagator and is incompatible with any completely positive trace-preserving (CPTP) reduced dynamics. We examine three objects with different causal structure under projection -- the Nakajima-Zwanzig (NZ) memory kernel $\tilde{\mathcal{K}}(z)$, the reduced-state Laplace transform $\tilde{\sigma}(z)$, and the force-fit effective kernel $\tilde{\mathcal{K}}_\text{eff}(z)$ -- using the Kramers-Kronig (KK) relations as diagnostic. Under a real-axis spectral hypothesis on the projected generator, $\tilde{\mathcal{K}}(z)$ lies in a vector-valued Hardy space and obeys (subtracted) KK relations, giving a CPTP-consistency criterion, a passivity-analyticity statement, and a Carleman diagnostic. We prove $\tilde{\sigma}(z)$ analytic in the upper half-plane for any initial state -- unitarity bounds $\|\sigma(t)\|_{\mathrm{op}} \leq 1$, so acausality cannot be blamed on the state alone. Yet the force-fit kernel can develop upper-half-plane poles at simple zeros of $\tilde{\sigma}(z)$: passive baths sit in a robust regime where these zeros stay real, while near-resonant systems enter a fragile regime in which coherence-channel zeros migrate into the upper half-plane, an intrinsic symmetry property already present for factorized states. We verify the full operator-valued KK relation on the extracted $4\times4$ NZ memory kernel of the Jaynes-Cummings model, the relative $L_2$ residual decreasing under refinement ($3.8\% \to 0.95\%$), consistent with exact matrix-valued KK in the continuum limit.

In this paper, we formulate the Schrodinger equation in gauge-theoretic terms. Starting from the Madelung representation, we rewrite the conserved probability current using gauge fields, namely a one-form gauge field in the $(2+1)$-dimensional theory and a two-form gauge field in the $(3+1)$-dimensional theory. This gives a local equivalence between the Schrodinger equation, quantum hydrodynamics and a non-relativistic gauge theory, while the global information is carried by the quantization condition of phase winding around zeros of the wavefunction. We then use this correspondence to study how topological deformations of gauge action and symmetry properties are represented in the wavefunction and fluid descriptions. On the gauge side, BF couplings to additional one-forms account for electromagnetic coupling, Berry connections, spinor dynamics, adiabatic non-abelian Berry connections, and intrinsic holonomy. Chern-Simons term admit, after eliminating the gauge field, a nonlocal realization in terms of wavefunction. This functional retain the topological content of the gauge description, but also contain dynamical contribution. In the presence of boundaries, the topological terms produce edge degrees of freedom and boundary charge algebras. Finally, in the nonlinear regime with a Bogoliubov sound mode, the dual two-form description relates acoustic memory to large gauge transformations and identifies the soft sector expected to complete the corresponding infrared triangle.

We study the relation between the Bogoliubov-de Gennes equation and the Ginzburg-Landau equation for a BCS model without external fields. While previous rigorous derivations of Ginzburg-Landau theory from BCS theory have focused on energies and minimizers, here we consider arbitrary critical points in the relevant energy regime. For temperatures close to the critical temperature, we prove that every sufficiently small solution of the BdG equation admits an asymptotic factorization into a microscopic Cooper-pair profile and a macroscopic order parameter. The latter satisfies the Ginzburg-Landau equation up to an error that vanishes in the scaling limit. Our analysis relies on a Birman-Schwinger reformulation of the BdG equation, a Lyapunov-Schmidt type reduction, and semiclassical estimates at low regularity.

We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking--Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu--Wu--Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p < 1$.

We consider a wide class of $2D$ tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant ${\cal N}_3$ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in $2D$, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the $K^{-1}$ group (generated by the Green function) with the specific element of the cyclic cohomology group $HC^3$. According to a set of local index theorems the values of ${\cal N}_3$ can be shown to be integer for a limited class of tight - binding models.

We prove a local bijection between alternating sign matrices (ASMs) and descending plane partitions (DPPs) in the refined fibres of Behrend--Di Francesco--Zinn-Justin. The map is constructed by moving a boundary line through the ASM and DPP pictures. At each step only a bounded neighbourhood of that boundary is changed. In growth-diagram language, the boundary states are ordinary ASM row frontiers and shifted DPP path frontiers, and the elementary moves are reversible Fomin-type square rules. The proof checks that overlapping local moves are confluent, that every square is invertible, and that the completed boundary again satisfies the ordinary ASM and DPP coordinate inequalities. The only terminal choice is resolved by a radius-two fan rule on the visible height contour. The same local framework gives a statistic on refined ASMs whose generating function is the DPP degree enumerator. The statistic has a Striker--Fulmek quadratic core \[ Q_{\rm SF}(A)= \sum_{1\le r<s\le n}\ \sum_{1\le b<a\le n} (n-r+1)A_{r,a}A_{s,b}, \] plus a bounded fan-boundary correction and, outside the normal boundary gauge, an explicit compact-potential endpoint term. In the normal gauge this is \[ W_{\rm ASM}^{\rm SF}(A)=Q_{\rm SF}(A)+B_{\rm fan}(A). \] Thus the paper gives both a local ASM--DPP bijection and a statistic-preserving refined \(q\)-enumeration. The supplementary files reproduce selected local checks and the displayed arithmetic examples.

Starting with a coupled discrete reaction--diffusion formulation for the lit and latent order books with non-uniformly sampled event times and meta-order source terms we show how two familiar market-microstructure regularities can emerge from this framework: the long-memory of trade signs associated with the Lillo--Mike--Farmer (LMF) theory and the square-root law (SQRL) of meta-order impact. This uses the locally linear order book and constant participation-rate execution in the front dynamics to reduce the dynamics to a Volterra equation whose leading-order solution then yields the well know result of concave impact trajectory, and a completion impact proportional to the square root of the meta-order size. We then use the interface representation to show how heavy-tailed Pareto meta-order lengths generate power-law trade-sign autocorrelations through the source term. These are familiar derivations, what is slightly different here is that we reinterpret these known derivations to make it clear that LMF law is an event-time sign-memory statement, whereas the square-root law is a physical-time viability statement where subordination can alter the calendar-time impact trajectories depending on the mappings and interpolation used to set continuum operational time.

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie--Bohm representation, in which stationary amplitude current constraints generate separable Sturm--Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie--Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie--Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm--Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

Color graphs and their subgraphs, referred to as bubble graphs, correspond bijectively to the simplicial complexes of pseudomanifolds and their subsimplices, respectively. In this paper, we introduce matrix representations for colored graphs and their associated bubble graphs. By using this correspondence, we define simplicial-complex matrices and subsimplex matrices that encode the simplicial complexes of pseudomanifolds and their subsimplices. Moreover, we formulate mutation and crossover operations on colored graphs. Through the established correspondence among simplicial complexes, colored graphs, and simplicial-complex matrices, we extend these operations to simplicial complexes and simplicial-complex matrices. We further implement an algorithm generating simplicial-complex matrices and a genetic algorithm performing mutation and crossover of them to produce pseudomanifolds exhibiting diverse topologies. In addition, we implement procedures for decomposing the generated simplicial-complex matrices into simplex matrices, reconstructing the simplicial complexes of the associated pseudomanifolds from this information, and computing geometric quantities such as the volume, circumcenter, and dual-simplex volume of each simplex.

The Bures--Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three--sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton--DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function $\Psi=\Phi^2$ satisfies the linear forced heat equation \begin{equation*} D_t\Psi=\Psi_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures--Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature $6/(1-4t)$ and extinction time $T=1/4$. Thus the metric remains inside the monotone cone for all $t<T$ and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume--normalized flow, for which the Bures--Helstrom metric is a fixed point. Its linearization is the shifted round--sphere Laplacian $\Delta_{\mathbb S^3}+3$, with spectrum \begin{equation*} \sigma_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.