A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3\Sigma$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

Most of the known non-invertible symmetries of quantum field theories in three and four spacetime dimensions act invertibly on local operators. An exception is coset symmetries, which can be constructed from gauging a non-normal subgroup of an invertible symmetry. In this paper, we study the action of a general finite non-invertible symmetry on local operators in four dimensions. We show that non-invertible symmetries without topological line operators necessarily act invertibly on local operators. Using this result, we argue that the action of a general non-invertible symmetry in 3+1d on local operators can be decomposed into the invertible action of some operators composed with the action of a gauging interface. We use this result to study when such a symmetry is anomaly-free. We find a necessary condition for a finite non-invertible symmetry in 3+1d to be anomaly-free, and show that anomaly-free non-invertible symmetries without topological line operators are non-intrinsically non-invertible.

We introduce a spectral approach to characterizing the three-dimensional Edwards-Anderson spin glass. By analyzing the eigenvalue statistics of overlap matrices constructed from two-dimensional cross-sections, we identify a crossover from the Wigner semicircle law at high temperatures towards a Gaussian distribution, which is consistently attained near the spin-glass critical point. Visible for different distributions of the random coupling, the Gaussian distribution can potentially serve as a robust spectral indicator of criticality. Remarkably, the spectral density is well-described by Tsallis statistics, with the entropic index $q$ evolving from $q = -1$ (semicircle, $T=\infty$) to $q = 1$ (Gaussian) at $T_c$, revealing a statistical structure inside the paramagnetic phase. We find $q\le 1$ within numerical precision. While the local level statistics remain consistent with GOE statistics, reflecting standard level repulsion, the temperature dependence appears mainly in the global spectral density. Our results present spectral statistics as a computationally efficient complement to multi-replica correlator methods and provide a new perspective on cooperative and critical phenomena in disordered systems.

We undertake a detailed analysis of a reaction-advection-diffusion (RAD) equation from the viewpoint of pulse-response studies, with particular attention to effects due to the advection velocity. Our boundary-value problem is a mathematical model for a system consisting of a narrow reactor tube into which a short pulse of reactant gas is injected at one end and a mixture of reaction product and unreacted gas flows out at the opposite end. Exit flow properties such as moments and peak characteristics are obtained analytically as functions of the Péclet number. The description of a standard transport curve\ -- -including diffusion and advection but no reaction\ -- -can serve as the baseline for further characterization of chemical activity. This characterization is done here for a first order irreversible reaction. Among our main observations is that chemical activity is easily obtained from the ratio of the exit flow curve in the presence of reaction over the standard transport curve.

Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.

We investigate the Maximal Entropy Simple Symmetric Exclusion Process (MESSEP) on a discrete ring with L sites and N indistinguishable particles. Its eigenfunctions are Schur polynomials evaluated at the L-th roots of unity, yielding an explicit spectral decomposition. The analysis relies on this eigenstructure and on the link between Schur polynomials and irreducible characters of the symmetric group, which forms the core algebraic tool for the scaling limits. In the low-density regime, where N is fixed and L tends to infinity, the rescaled dynamics converge to the Unitary Dyson Brownian Motion (UDBM). The electrostatic repulsion then emerges as an entropic force, providing a canonical microscopic derivation of the UDBM. In the hydrodynamic regime, where N is equivalent to $\alpha$L with $\alpha$ P p0, 1q, the empirical measure converges to a density solving a nonlinear, nonlocal transport equation. Its moment generating function satisfies a complex Burgers-type equation. As $\alpha$ tends to 0, this equation coincides with that governing the spectral distribution of the Free Unitary Brownian Motion (FUBM), thereby bridging discrete entropic exclusion dynamics and free unitary hydrodynamics. Overall, the MESSEP provides a unified canonical discrete framework connecting unitary Dyson motion and free unitary Brownian motion through nonlinear hydrodynamic limits, with Schur and character theory as the central algebraic structure.

In stellarator design, the coil winding surfaces $\Sigma\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.

We construct nonlinear wave operators and prove small-data asymptotic completeness for the Maxwell--Higgs system on the domain of outer communications of every four-dimensional subextremal Kerr black hole $(\mathcal D_{M,a},g_{M,a})$ with $M>0$ and $|a|<M$, for gauge-invariant nonnegative scalar potentials $P$ satisfying Assumption~\ref{asumsiP} with mass parameter $m^{2}\ge0$. The massless case $m=0$ is unconditional on the full subextremal range. For $m^{2}>0$ the same conclusions follow assuming the massive linear package $\Lin_{k}^{(m)}$ for the linear comparison system (in particular, no exponentially growing modes); this fails for an open set of masses due to superradiant instability \cite{ShlapentokhRothmanKGKerr}. We work in the radiative (charge-free) regime; stationary Coulomb (Kerr--Newman) modes are treated separately. Asymptotic states are described by gauge-covariant radiation fields on $\mathcal I^{\pm}\cup\mathcal H^{\pm}$ (and, when $m>0$, an additional timelike/Dollard channel), yielding a gauge-invariant nonlinear scattering map on the residual-gauge quotient. The scattering map is a small-data bijection, is Fréchet differentiable at $0$ with derivative equal to linear Kerr scattering, admits a quadratic (Born) expansion with an $O(\|U\|^{3})$ remainder in the natural asymptotic topology, and is real-analytic for analytic $P$. The nonlinear argument is presented as a transfer principle from a black-box linear estimate package for inhomogeneous Klein--Gordon and charge-free Maxwell fields, verified here in the massless Kerr case (and proved self-contained in Schwarzschild).

The interfacial diffusion associated with finite volume method (FVM) discretizations of multiphase flows creates the need for an interface sharpening mechanism. Such solutions for structured quadrilateral grids are well documented, but various engineering applications require mesh designs specific to the irregular geometry of the physical system it is modeling. Therefore this study casts interface sharpening as an ant-idiffusive volumetric body force whose calculation procedure is generalizable to an arbitrarily constructed grid. The force magnitude is derived at cell centers as a function of the local compressible flow characteristics and the geometry of the cell neighborhood. The flow model uses an AUSM+up based method for flux evaluation and imposes a stiffened equation of state onto each of the fluids in order to close the linear system and extract auxiliary variables. Validation tests show good agreement with the Young-Laplace condition whereby the interface converges to the analytical solution corresponding to a balance between a pressure jump and interfacial forces. Further results show the recovery of a circle starting from a shape with highly variational curvature through the combined effects of surface tension and interface sharpening. Lastly shear-driven droplet pinchoff results show good agreement with droplet shapes provided by the surrounding literature at various Weber-Ohnesorge number combinations.

We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's function is the Gaussian wave. This generalizes a result of Backhausz and Szegedy, who proved this for the infinite regular tree of degree $d\geq 3$. We do this by giving a reduction to a statement concerning the distribution of the inner product of our process with columns of the Green's function, which in turn are straightforward to calculate. As a consequence, for random bipartite biregular graphs, the distribution of local neighborhoods of eigenvectors must approximate the Gaussian wave. Moreover, for generic configuration models including random lifts, the local distribution of a uniformly chosen eigenvector from any arbitrarily small spectral window likewise converges to the Gaussian wave.

We study relative differential and integral forms on families of supermanifolds and their cohomology. We prove a relative Poincaré--Verdier duality and show that it relates the cohomology of differential and integral forms, admitting a concrete geometric realization via Berezin fiber integration. We further introduce the Poincaré--dual integral form associated to an embedded even family and prove that it satisfies the correct localization property. We then apply these results to supergravity, focusing on the $3d$ case. In this setting, we show that relative Poincaré duality provides the natural framework for encoding the data needed to relate a superspace formulation to the physical spacetime, thereby yielding a rigorous definition of picture changing operators used in the physics literature. Building on this, after a careful analysis of the space of fields and the relevant constraints, we prove that the component, superspace, and geometric formulation of the theory are all equivalent. Finally, under suitable hypotheses, we argue that our construction illustrates a general principle governing the mathematical formulation of classical field theories on supermanifolds.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $\rho$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $\rho$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category $\mathcal{C}$, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center $Z(\mathcal{C})$. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.

From any graph $G$ arises a flat space wavefunction, obtained by integrating a product of propagators associated to the vertices and edges of $G$. This function is a key ingredient in the computation of cosmological correlators, and several representations for it have been proposed. We formulate three such representations and prove their correctness. In particular, we show that the flat space wavefunction can be read off from the canonical form of the cosmological polytope, and we settle a conjecture of Fevola, Pimentel, Sattelberger, and Westerdijk regarding a partial fraction decomposition for the flat space wavefunction. The terms of the decomposition correspond to certain collections of connected subgraphs associated to $G$ and its spanning subgraphs, reflecting the fact that the flat space wavefunction contains information about how $G$ is connected.

In this paper, we prove that biorthogonal ensembles on the real line with a specific derivative structure admit an explicit correlation kernel of double contour integral form. We will demonstrate that this expression is a valuable starting point for asymptotic analysis and that our class of biorthogonal ensembles admits a large variety of limit kernels, by proving that two new classes of limit kernels can occur. The first type is a deformation of the hard edge Bessel kernel which arises in polynomial ensembles describing the eigenvalues of the sum of two random matrices, while the second type arises for Muttalib-Borodin type deformations of polynomial ensembles.

Insufficient reference data is ubiquitous in data-driven computational fluid dynamics, as it is usually too expensive to compute or impossible to observe over long enough times needed for data-driven methods. The lack of data can significantly compromise the fidelity of results computed with data-driven methods or render them inapplicable. To challenge this problem, we propose a probabilistic reconstruction method that enhances the hyper-parameterisation (HP) approach with ideas underlying the probabilistic-evolutionary approach. We offer to use the HP method ``Advection of the image point'' on data sampled from the joint probability distribution of the reference dataset. The HP method has been tested regionally on the sea surface temperature and surface relative vorticity computed with the global 1/4-deg and 1/12-deg resolution NEMO model. Our results show that the HP solution (the solution computed with the HP method) in the probabilistically-reconstructed and reduced (in terms of dimensionality) phase space at 1/4-deg resolution is more accurate than the 1/4-deg-solution computed with NEMO. Additionally, the HP solution is several orders of magnitude faster to compute than the 1/4-deg NEMO solution. The proposed method shows encouraging results for the NEMO model and the potential for the use in other operational ocean and ocean-atmospheric models for both deterministic and probabilistic predictions. Furthermore, the method can be used as a fast reanalysis tool allowing the complex dynamics of a comprehensive ocean model to be approximated by the HP solution. It can also function as a dynamic interpolation method to fill gaps in observational data.

Spherical spin random fields are used to model the Cosmic Microwave Background polarization, the study of which is at the heart of modern Cosmology and will be the subject of the LITEBIRD mission, in the 2030s. Its scope is to collect datas to test the theoretical predictions of the Cosmic Inflation model. In particular, the Minkowski functionals, or the Lipschitz-Killing curvatures, of excursion sets can be used to detect deviations from Gaussianity and anisotropies of random fields, being fine descriptors of their geometry and topology. In this paper we give an explicit, non-asymptotic, formula for the expectation of the Lipshitz-Killing curvatures of the excursion set of the real part of an arbitrary left-invariant Gaussian spin spherical random field, seen as a field on $SO(3)$. Our findings are coherent with the asymptotic ones presented in Carrón Duque, J. et al. "Minkowski Functionals in $SO(3)$ for the spin-2 CMB polarisation field", Journal of Cosmology and Astroparticle Physics (2024). We also give explicit expressions for the Adler-Taylor metric, and its curvature. We obtain such result as an application of a general formula that applies to any nondegenerate Gaussian random field defined on an arbitrary three dimensional compact Riemannian manifold. The novelty is that the Lipschitz-Killing curvatures are computed with respect to an arbitrary metric, possibly different than the Adler-Taylor metric of the field.

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.

Quantum state exclusion is an operational task with application to ontological interpretations of quantum states. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, admitting the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, when there are two hypotheses, our result provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, when all channels are classical, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion.

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

We study the resurgent structure of the topological string dual to 2d $U(N)$ Yang-Mills on torus. We find closed form formulas for instanton amplitudes up to arbitrarily high instanton orders, based on which we propose the non-perturbative partition function including contributions from all the real instantons, which is real for positive modulus and string coupling. We also explore complex instantons and find two infinite towers of them. We expect them to correspond to BPS states in type II string.

Quantum conditional entropies play a fundamental role in quantum information theory. In quantum key distribution, they are exploited to obtain reliable lower bounds on the secret-key rates in the finite-size regime, against collective attacks and coherent attacks under suitable assumptions. Here we consider continuous-variable communication protocols, where the sender Alice encodes information using a discrete modulation of phase-shifted coherent states, and the receiver Bob decodes by homodyne or heterodyne detection. We compute the Petz-Rényi and sandwiched Rényi conditional entropies associated with these setups, assuming either a passive eavesdropper or one that injects thermal photons into the channel, who gathers the quantum information leaked through a lossy communication line of known or bounded transmittance. Whereas our results do not directly provide reliable key-rate estimates, they do represent useful ball-park figures. We obtain analytical or semi-analytical expressions that do not require intensive numerical calculations. These expressions serve as bounds on the key rates that may be tight in certain scenarios. We compare different estimates, including known bounds that have already appeared in the literature and new bounds. The latter are found to be tighter for very short block sizes.

We investigate the existence of generalised densities for the $\Phi^4_d$ $(d=1,2,3)$ measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the $\Phi^4_1$ measure coincides with the $\Phi^4$ action as expected. In two dimensions, we show that OM functionals of the $P(\Phi)_2$ measures agree with the corresponding actions, by considering ``enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension $3$, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the $\Phi^4_3$ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

Lubrication theory makes use of the assumptions of a long and thin fluid domain and a small scaled Reynolds number to formulate a linearized approximation to the Navier-Stokes equations. Extended lubrication theory aims to improve the model accuracy by relaxing these assumptions and including additional terms in the formulation. However, such models are sensitive to large surface gradients which lead the assumptions of the model to break down. In this paper, we present a formulation of extended lubrication theory, and compare our model with several existing models, along with the numerical solution to the Stokes equations. The error in pressure and velocity is characterized for a variety of fluid domain geometries. Our results indicate that the new solution is suitable for a wide range of geometries. The magnitude of surface variation and the length scale ratio are both important factors influencing the accuracy of the extended lubrication theory models.

The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Grönwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase in non-Hermitian systems.

We establish the integrability of a family of Sachdev-Ye-Kitaev (SYK) models with uniform $p$-body interactions. We derive the R-matrix and mutually commuting transfer matrices that generate the Hamiltonians of these models, and obtain their exact eigenspectra and eigenstates. Remarkably, the R-matrix is that of the critical transverse-field Ising chain. This work reveals an unexpected connection between the SYK model, central to many-body quantum chaos, and the critical Ising chain, a cornerstone of statistical mechanics.

Boundary-catalytic branching processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on absorbing regions of the boundary. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction in a bounded domain. In other words, we establish a powerful tool for calculating the growth-regulating absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

There exist several different proposals for a measure in Quantum Gravity theories. Although sometimes being labelled as non covariant, the measure derived in [7] for GR has the particularity that, in the extremal, the volume divergences cancel out. The analogous for Quadratic Gravity [1]-[2] was considered in [44]-[45]. However, as far as the author knows, the issue of volume divergences was not considered for this last measure. The present work fills this gap and presents an analysis showing that, in the extremal, these divergences cancel as well. This is up to some subtleties related to superdeterminants. The possibility of employing non invariant measures may be accepted if the anomaly in the measure is compensated by counter term redefinitions of the model under analysis. This makes difficult to disprove, at the present times, some choices of measures. Quadratic Gravity[1]-[2], is known to be renormalizable in flat space, and there are a finite number of counter terms needed in order to renormalize its effective action. However, around a curved space this is not known, and this complicates considerably the analysis. These issues are reviewed in the text, together with an analysis of covariant measures. In particular, it is shown how these measures [47]-[49] can be found if one condition in [7] is relaxed.

Building on the earlier work by Araki and Tanii, Aschieri et al., and Buratti et al., we demonstrate that every model for self-dual nonlinear electrodynamics in four dimensions has a $\mathsf{U}(1)$ duality-invariant extension to $4p>4$ dimensions and construct new self-dual nonlinear theories for a gauge $(2p-1)$-form. We present a family of models for self-dual $(2p-1)$-form electrodynamics in which the trace of the energy-momentum tensor determines the flow with respect to a duality-invariant deformation parameter.

The Reynolds equation is derived from the incompressible Navier Stokes equations under the lubrication assumptions of a long and thin domain geometry and a small scaled Reynolds number. The Reynolds equation is an elliptic differential equation and a dramatic simplification from the governing equations. When the fluid domain is piecewise linear, the Reynolds equation has an exact solution that we formulate by coupling the exact solutions of each piecewise component. We consider a formulation specifically for piecewise constant heights, and a more general formulation for piecewise linear heights; in both cases the linear system is inverted using the Schur complement. These methods can also be applied in the case of non-linear heights by approximating the height as piecewise constant or piecewise linear, in which case the methods achieve second order accuracy. We assess the time complexity of the two methods, and determine that the method for piecewise linear heights is linear time for the number of piecewise components. As an application of these methods, we explore the limits of validity for lubrication theory by comparing the solutions of the Reynolds and the Stokes equations for a variety of linear and non-linear textured slider geometries.

The Lorentz mirror model provides a clean setting to study macroscopic transport generated solely by quenched environmental randomness. We introduce a hierarchical version whose distribution of left--right crossings satisfies an exact recursion. In dimensions $d\geq 3$, we prove normal transport: the mean conductance scales as (cross-section)/(length) on all length scales. A Gaussian closure, supported by numerics, predicts that the variance-to-mean ratio of the conductance converges to the universal value $2/3$ for all $d\geq 2$ (the ``$2/3$ law''). We provide numerical evidence for the $2/3$ law in the original (non-hierarchical) Lorentz mirror model in $d=3$, and conjecture that it is a universal signature of normal transport induced by random current matching. In the marginal case $d=2$, our hierarchical recursion reproduces the known scaling of the mean conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: this https URL

A.~Einstein considered a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric and $F\ne0$ is skew-symmetric, and a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. M. Prvanović (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the $f^2$-torsion condition $T(f^2X, Y) = T(X,f^2Y) = f^2 T(X,Y)$, where $F(X,Y)=g(X,fY)$. We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds $(M,f,g)$, satisfying the $f^2$-torsion condition, give explicit formulas for the torsion in terms of $\nabla^g F$, $dF$ and a new (1,1)-tensor $\widetilde Q=-f^2-{I}$, and show that in the almost Hermitian case, our results reduce to the coordinate-free form of Prvanović's solution. Finally, we discuss special Einstein connections, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.

We present the exact analytic solution of the Makeenko-Migdal loop equations, solving planar QCD ($N_c \to \infty$) in the continuum limit. Quantizing internal Majorana fermions (elves) on a rigid Hodge-dual minimal surface provides the algebraic mechanism satisfying the unintegrated vector loop equations. The Pauli principle exactly cancels non-planar intersections to reproduce planar factorization, while iterating the equation generates the planar graphs of asymptotically free QCD. Holographically fixing the bulk geometry by the boundary loop strictly avoids Liouville instability. Momentum loop space integrates out coordinate-space cusp singularities, yielding a finite local limit. Gauge-fixing the Virasoro constraint parametrizes the reduced phase-space measure by boundary twistors. The theory reduces to a confining analytic twistor string: a boundary sigma model $S^1 \to (S^3 \times S^3)/S^1$ coupled to a holographically determined Liouville field. The meson spectrum becomes a 1D functional integral over boundary twistor trajectories. Complexified action monodromies reveal a discrete mass spectrum governed by Catastrophe Theory, classified by the topological number of twistor poles inside the unit circle. This geometric localization is an infinite-dimensional realization of Exact WKB analysis and Picard-Lefschetz resurgence, where twistor poles act identically to Seiberg-Witten branch points. The 1-pole sector yields the exact Regge spectrum $m^2 = \frac{\pi\sigma}{2}(n + 1/24)$, matching experimental $\pi, K, \rho$ trajectories within $95\%$ confidence. The empirically correct open-string intercepts emerge directly from the conformal anomaly of the microscopic elves rather than macroscopic string vibrations. Ultimately, this explicitly realizes Witten's Master Field as a critical classical trajectory in twistor space.