This work studies the asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region when the reflection coefficients associated with the initial data belong to weighted Sobolev space. The Dbar-steepest descent method is applied to the Riemann-Hilbert problem and the long-time asymptotic expansion of the solution are obtained up to an optimal error of order $\mathcal{O}(t^{-3/4})$. Compared with previous results, we extend the initial data from the rapidly decaying Schwartz space to a weighted Sobolev space, and prove the asymptotic stability of the solution in dispersive wave region.

As fusion energy moves from theoretical feasibility toward commercialization, design of new reactor concepts, autonomous tokamak control, and high-performance scenario optimization are becoming increasingly important. Traditionally, such optimization tasks have relied on costly trial-and-error or brute-force parameter searches, based on black-box experiments or simulations. Recently, advances in differentiable programming are changing the paradigm of numerical simulation. Unlike conventional simulations, which are typically executed as locally connected step-by-step procedures, differentiable simulation represents the entire simulation pipeline as a connected computational graph. In such a framework, machine parameters, actuator waveforms, and plasma responses are linked through differentiable operations, allowing Jacobians to propagate across the full simulation. This enables direct gradient-based control and optimization using the internal sensitivities of the simulator, rather than treating the simulator as a black box. Here, we present TokaGrad, an end-to-end differentiable tokamak transport simulator for full-scenario modeling, including ramp-up, L-mode operation, and H-mode access. TokaGrad self-consistently integrates differentiable models for plasma equilibrium, transport, heating, L-H transition, and pedestal formation. To our knowledge, this is the first differentiable tokamak simulator capable of self-consistently modeling dynamic full-discharge scenarios where actuators and plasma evolve together with equilibrium, pedestal, and confinement-regime transitions. We demonstrate that, when coupled to gradient-based optimizers, TokaGrad enables reactor-design optimization, actuator control, and full-scenario waveform optimization. This framework provides a pathway toward automated, differentiable optimization of burning-plasma scenarios and reactor concepts.

We study the entanglement entropy (EE) of ground states of a Hamiltonian defined on a domain with a boundary. Surprisingly, boundary conditions can change the spectrum and the nature of the spectrum drastically but not the leading behaviour of the EE. As is well-known, the Landau Hamiltonian on the full plane has pure point spectrum (the infinitely degenerate Landau levels) and ground states display a so-called strict area law. On the other hand, the Landau Hamiltonian on the half-plane has purely absolutely continuous spectrum and yet we prove a strict area law for its ground states. We raise the question of what extra or finer conditions on the absolutely continuous spectrum are necessary to guarantee a logarithmically enhanced area-law as we have for the Laplace operator.

We review and extend the results on the group analysis of a class of generalized Kawahara equations with time-dependent coefficients. First, we provide an overview of the existing literature on Lie symmetries and Lie-invariant solutions of such equations. We then present a complete description of the transformation properties of these equations, including admissible, equivalence, and Lie symmetry transformations. For practical applications, the results are further extended by presenting a complete Lie symmetry classification without simplifying the coefficients by equivalence transformations. The Lie reductions are then thoroughly performed and some Lie exact solutions are constructed. The local conservation laws of low order are classified: every equation from the class admits the mass and the $L^2$-norm conservation laws, whereas energy-type conservation laws exist only for distinguished coefficient branches that agree with the cases singled out by the symmetry classification. The classification results are enhanced by a study of contractions, which link cases of Lie symmetry extensions together with the associated reductions and conservation laws.

Vertex operator algebras (VOAs) are well studied in both mathematics and physics. The best understood class is that of strongly rational VOAs, whose representation category is maximally well behaved: indeed, it is a modular tensor category. At the next level of complexity are strongly finite but non-rational VOAs. Their representation category is not semisimple (it is ``logarithmic''), but maintains nice structural properties. Only a few families of examples in this class are known, a fact that may have hindered the development of a comprehensive mathematical theory. The SCFT/VOA correspondence provides a natural way to generate more examples: strongly finite but non-rational VOAs are expected to arise from four-dimensional ${\cal N}=2$ Lagrangian superconformal field theories (SCFTs) that do not admit a Higgs branch moduli space of vacua. We tackle the combinatorial task of classifying all such ``Higgsless'' Lagrangian SCFTs. To our surprise, this set turns out to be rather sparse. Free vector multiplets and their discrete gaugings are immediate examples. The interacting Higgsless theories comprise one infinite sequence of SO/USp quivers and three sporadic examples. We construct and study the novel VOAs associated to two of the sporadic examples, and confirm that they are indeed strongly finite and logarithmic.

We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four $\sigma$ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-$\sigma$ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply $\Pr_L(\mathrm{fail})\le C|\Omega_L|e^{-cL}$ below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.

We formalize a research result in the Lean 4 proof assistant by having a mathematician direct an AI system, and frame the activity as a formalization game. The objective is to turn a LaTeX document into Lean. The game is won when the development compiles, contains no sorry, and a machine check shows the target theorems rest on Lean's foundational axioms alone. Reuse is a second check, by a definition we introduce: whether the development yields a self-contained layer of general mathematics the wider library could absorb. The case study is a complete, axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin's mean-field route -- existence, uniqueness, the stability estimate and mean-field limit, and a short-window superposition principle (weak solutions are Lagrangian). The human's role was to direct, not to write proofs: to scope the definitions, steer the decompositions, and triage the library's gaps; the AI agent executed. The formalization certifies the proof of each statement as written; whether the written statement is the intended theorem stays the mathematician's judgment. The optimal-transport machinery that fell out of the build (in particular, properties of the Wasserstein-1 metric and the Kantorovich-Rubinstein duality theorem) separates into a self-contained layer that compiles against Mathlib alone: about a sixth of the development (49 of 299 declarations), behind a 22-declaration interface with no reverse dependency. The headline theorems ran in about a week, the full development in about a month. We report the quantitative claims as observations of one game, not as general laws. The game's rules name no particular system, so the methodological framing is meant to outlast the tools of any one run.

Synchronization is ubiquitous across natural and synthetic systems, yet most prior studies focus on the inertia-free Kuramoto model and do so at the macroscopic level. In this study, we instead investigate the inertial Kuramoto model and analyze the kinetics of individual synchronized clusters that emerge in the underdamped dynamics, driven by the interactions among multiple synchronized clusters with different frequencies. Specifically, we explore two forms of intrinsic frequency distribution -- unimodal Gaussian and multimodal uniform -- and show that they give rise to qualitatively different synchronized clusters: a hierarchical organization for the Gaussian distribution and a homogeneous organization for the uniform distribution. This contrast leads to qualitatively different behaviors of the order parameter: for the Gaussian distribution, it increases smoothly with increasing coupling strength, while for the uniform distribution, it grows through a series of discrete jumps that trace out the size of the Devil's staircase (DS). By resolving the kinetics at the cluster level, we further find that the route to synchronization also depends on the distribution type: with a Gaussian distribution, a single dominant cluster forms and gradually entrains the remaining oscillators, whereas with a uniform distribution, synchronization proceeds via successive cluster mergers initiated from peripheral seeds associated with the high-frequency periphery. Taken together, these findings provide a new perspective on collective synchronization dynamics in inertial complex systems.

We show that the dispersionless version of the modified DKP hierarchy originally defined as the limit of relations for the tau-function of the Hirota-Miwa type has an equivalent reformulation as the Yang-Baxter equation for Baxter's $R$-matrix of Boltzmann weights for the 8-vertex model.

Symmetric Macdonald polynomials of $n$ variables provide eigenfunctions of the $N$-body trigonometric Ruijsenaars-Schneider integrable system at particular eigenvalues. In order to construct eigenfunctions with arbitrary eigenvalues, M. Noumi and J. Shiraishi used a recursion in $N$ (branching rule) for the symmetric Macdonald polynomials and analytically continued them. This generated a power series, which is a part of triad (universal solution). In the present paper, we demonstrate that a similar procedure is available for another integrable system, $N$-body Cherednik system inspired by the DAHA of type $A$, which has non-symmetric Macdonald polynomials as its polynomial eigenfunctions. However, in this system, the generic eigenfunction is more complicated: it is not just a simple power series as in the Noumi-Shiraishi case, but has an involved structure with $N!$ branches, each of them being a power series of the Noumi-Shiraishi type. As an illustration, we also provide explicit formulas for particular cases.

We consider the Cauchy problem for the nonlinear Dirac equation on a noncompact $N$-star metric graph $G$, \[ \mathrm{i}\partial_t \psi = D\psi - |\psi|^{p-2}\psi, \qquad \psi(0)=\psi_0, \] where $p\ge3$, $\psi:\mathbb{R}\times G\to\mathbb{C}^2$ and $D$ denotes the self-adjoint Dirac-Kirchhoff operator on $G$. Using Bourgain-type spaces defined through the spectral resolution of $D$, together with elementary $L^\infty$ bounds for the Dirac flow and fractional Nemytskii estimates below the trace threshold, we prove local well-posedness for initial data \[ \psi_0\in H_D^s(G)\cap L^\infty(G;\mathbb C^2), \qquad 0\le s<\frac12 . \] The corresponding solution belongs to \[ C([0,T];H_D^s(G))\cap X_T^{s,b}\cap L^\infty([0,T]\times G). \] Moreover, $\|\psi(t)\|_{L^2(G;\mathbb{C}^2)}$ is conserved along the solution on the existence interval. We also establish a blow-up alternative in the combined $H_D^s$ and space-time $L^\infty$ control norm.

The Einstein-de Sitter universe is the prevailing model used in cosmology to describe the cold dark matter-dominated epoch of the universe. This model is a spatially homogeneous and isotropic spacetime undergoing decelerated expansion, and is linearly unstable under the Einstein-Euler equations with a pressureless fluid equation of state. We show that every initial data set for the Einstein-Euler equations on $\mathbb{T}^3$ with a near-flat metric and positive fluid energy density converges to a flat metric under the Einstein-Euler flow with a polytropic equation of state. This means the metric asymptotes to an Einstein-de Sitter spacetime. In particular, this settles the question of whether the Einstein-de Sitter model can be nonlinearly stable for an appropriate matter model.

We develop a rigid $\mathcal{N}=(1,1)$ superspace formulation for multifield scalar models coupled to localized impurities through spurion superfields in two-dimensional spacetime. The spurionic completion gives a manifestly supersymmetric action and provides a systematic framework for describing interacting scalar fields in the presence of impurity backgrounds. In this setting, supersymmetry acts as an organizing principle for a controlled bosonic half-BPS sector, with the preserved projector selecting the impurity-compatible first-order equations. We derive the corresponding coupled BPS equations, energy density, boundary conditions, and Bogomol'nyi bound, showing that localized impurities deform the BPS profiles and redistribute the local energy density while leaving the total BPS energy fixed by the topological boundary term. We illustrate the formalism through representative one-, two-, and three-field models, analyzing the resulting BPS configurations and their local energy density profiles.

We formulate a generally covariant six-field Rational Extended Thermodynamics model (RET$_6$) for relativistic polyatomic gases, with the dynamical pressure as the only non-equilibrium variable. The model is based on a polyatomic extension of the Boltzmann-Chernikov kinetic equation, where the one-particle distribution depends also on an internal-energy variable, and on the Maximum Entropy closure of the associated relativistic moment hierarchy. The resulting field equations, closure relations, and production term are therefore fixed by the underlying kinetic structure rather than postulated phenomenologically. We extend the RET$_6$ model from Minkowski spacetime to a general curved spacetime by the minimal coupling prescription and couple it to the Einstein equations. As a first structural result, we prove a kinetic-theory no-go theorem in this polyatomic RET setting stating that any stress-energy tensor induced by a non-negative relativistic one-particle distribution function satisfies the strong energy condition. We then specialize the theory to a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. In this setting the dynamical pressure modifies the expansion dynamics with respect to the perfect-fluid Euler case, but the no-go theorem excludes acceleration driven by the RET$_6$ gas alone. Finally, we reintroduce a cosmological constant and study the combined $\Lambda$RET$_6$ model, proving the existence and local stability of a de Sitter attractor at late times. Numerical integrations show that, for physically motivated post-recombination initial data and relaxation times, the expansion history rapidly approaches that of $\Lambda$CDM, with small non-equilibrium corrections controlled by the relaxation time and by the initial value of the dynamical pressure.

In this work we introduce and study a family of holomorphic--topological field theories, which we call generalized Poisson sigma models. These theories are higher-dimensional analogues of the two-dimensional Poisson sigma model, with target data encoded by shifted chiral Poisson structures. We investigate their relationship with deformation quantizations of holomorphic--topological factorization algebras. Along the way, we give a systematic construction of extended objects, including interfaces, enriched boundaries and defects based on relevant notions in derived algebraic geometry. We employ Koszul-duality methods to study boundary algebras, yielding various versions of quantum groups. We illustrate the general framework through a range of examples, including twists of supersymmetric gauge theories as well as examples beyond the supersymmetric origin.

The AdS/CFT correspondence relates gravity in anti-de Sitter space to a boundary conformal field theory, and in its AdS$_3$/CFT$_2$ instance the Virasoro symmetry of the boundary theory organizes correlation functions into conformal blocks. In the semiclassical limit these blocks are computed by lengths of geodesic networks in the bulk, most sharply in the heavy-light regime, where heavy operators source a background probed by light ones. We relate the classical monodromy method to this bulk geometry in holographic coordinates, showing that the eigenvectors of the monodromy matrix encode the endpoints of bulk geodesics. This yields the light action and the equations determining the internal geodesic network; crucially, the internal network equations are independent of the heavy background. For two heavy operators we rederive the same equations from elementary Euclidean geometry, which provides an independent geometric check. As an application we compute the full non-vacuum 5-point HHLLL block, so far known only in the superlight approximation. More broadly, our construction gives a general framework for computing heavy-light blocks from the bulk, while at the same time fixing its threshold of computability.

Kuznetsov and Sahinidis (J. Glob. Optim., 2025) prove distance bounds that confine optimal Lennard-Jones clusters and shrink the search region of deterministic solvers; their diameter bound charges each unit-width layer the loosest internal energy $-\binom{n}{2}$. We replace this by a certified estimate built from their own subset inequality and the proven minima $V_5^*$, $V_6^*$, and minimize the resulting layer bound over population profiles. Centred decreasing profiles supply candidate minima; an arrangement-free relaxation, whose only classical ingredient is the rearrangement inequality for sequences, closes every certificate over all profiles; and all comparisons are re-verified in directed-rounding arithmetic. For $5 \le N \le 200$ this certifies a one-layer improvement of the published bound at 92 sizes. The improvement resolves no open global-optimization case: it is a rigorous tightening of a published a priori bound, with a precise account of the mechanism. A direct downstream test on the only tractable sizes ($N \le 6$) finds the diameter box is not the binding resource for a deterministic solver there, and no solver runs at the sizes the refinement affects ($N \ge 38$); we therefore present the result as a theoretical note. We also derive the asymptotic form of the bound, $\rho_{KS} = N - \Theta(\sqrt{N})$, resolving a point left open by Kuznetsov and Sahinidis; the gain of the refinement itself grows like $\Theta(\sqrt{N})$ layers, with an exact asymptotic constant.

We present the H-cloud formalism, a cloud-accessible and open-source finite-element framework for electromagnetic modelling of applied superconductors. The proposed method expresses the nonlinear electromagnetic response of type-II superconductors in a curl-conforming discretisation based on Nédélec finite elements, where the tangential applied-field boundary condition, nonlinear E-J power law, and fully implicit time-discrete residual are stated explicitly at the variational level, all within a scripted Python finite-element workflow. The weak form is used as the basis for forward simulation and for extension to adjoint differentiation and PDE-constrained optimisation, while keeping the governing assumptions, boundary conditions, and solver structure fully visible to the user. The implementation is realised in Firedrake with UFL and PETSc-backed nonlinear solvers, allowing the identical script to run on local machines and in browser-accessible environments such as Google Colab without reformulating the problem. The method is verified on the canonical magnetisation benchmark of a cylindrical superconductor under Bean-like penetration conditions and then benchmarked against an independently constructed COMSOL model for a practical high temperature superconducting Bi2212 wire. Across matched mesh studies, the open-source workflow reproduces the commercial-reference magnetisation loops to within approximately \(1\%\) , with relative peak errors below 1.5%, while cloud execution preserves the same numerical solution at rather modest additional runtime considering the use of (freely available) reduced hardware resources. The proposed framework provides a rigorous, reproducible, and portable route for superconducting simulation, benchmarking, and future optimisation-led modelling of applied and functional superconductors, shareable and executable into open cloud environments.

This study investigates the boundary conditions for fluid-dynamic equations at the interface of a gas and its condensed phase during steady evaporation of a polyatomic gas. Evaporation curves illustrating the dependence of the temperature and pressure ratios on the Mach number are considered for a calorically perfect gas whose molecules behave like rigid rotors. At the condensed phase, complete absorption conditions are assumed. Also an extension to cases in which a part of the molecules is diffusely reflected at the condensed phase is also considered. We revisit the half-space evaporation problem for polyatomic gases based on previous results, applied to a slightly modified entropy functional. The structure of the (modified) maximal total entropy production curves is investigated, and simple functions that accurately fit the numerical results are proposed. Functions of the proposed form, with modified coefficients, also fit the evaporation curves previously obtained by different numerical methods surprisingly well. The approximation is performed for different numbers of internal degrees of freedom or ratios of specific heats. Simple functions that depend on the ratio of specific heats are found to fit the obtained evaporation curves for different ratios of specific heats very well.

Let $H_a=-\Delta+a|x|^{-2}$ be the Friedrichs extension on $L^2(\mathbb{R}^d)$, where $d\ge 3$ and $-(d-2)^2/4\le a<0$ lies in the attractive Hardy range. Starting from the known positive two-sided comparison for the kernel of $H_a^{-s/2}$, we determine the complete strong non-endpoint mapping range for two power weights. If $\sigma=(d-2-\sqrt{(d-2)^2+4a})/2$ and $0<s<d-2\sigma$, then [ ||x|^{-\beta}H_a^{-s/2}f|{L^q} \lesssim ||x|^\alpha f|{L^p} ] holds for $1<p,q<\infty$ precisely under the exponent ordering, scaling, sum, and origin conditions stated in the main theorem. At either origin-critical boundary, the strong estimate and the corresponding weighted $L^p\to L^{q,\infty}$ estimate fail, whereas the Lorentz replacement $L^{p,1}\to L^{q,\infty}$ holds. We also derive weighted Sobolev consequences and treat the Hardy-critical Friedrichs case separately. No new heat-kernel, spectral multiplier, Bernstein, or Littlewood--Paley theorem is claimed.

We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set $\Gamma$ of equivalence classes of $C_0$-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric $d$: the distance between two classes is the convergence exponent of the series $\sum_j|\langle\varphi_j,\psi_j\rangle-1|$ formed from any pair of representatives. We show that $d$ is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so $d$ separates points only after passing to the quotient $\widetilde\Gamma$ of $\Gamma$ by the relation $d=0$; the pair $(\widetilde\Gamma,d)$ is then a complete ultrametric space. As an application, we show that a product unitary $\bigotimes_j U$ whose factor $U$ satisfies $\inf_{\|x\|=1}|\langle x,Ux\rangle-1|>0$ (in particular, a unitary on a finite dimensional space with $1\notin\sigma(U)$) displaces every class to the maximal distance $1$. Guided by the intended application -- a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds -- we also develop a gauge-invariant variant $\tilde d$ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by $\tilde d$, is class dependent and realizes every value in $[0,1]$. We interpret $\tilde d$ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite $G$-symmetry. In particular, we argue that (3+1)d fermionic topological order with $G$-symmetry correspond to $\mathbf{2SVect}$-enriched $G$-crossed braided fusion 2-categories. We then show that the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang-Wen-Witten construction of bosonic topological theories with $G$-symmetry saturating an anomaly. More generally, we also explain how 2-categorical (de-) equivariantization yields a classification of all braided fusion 2-categories.

Computing the electronic structure of incommensurate materials is a central challenge in condensed matter physics, requiring efficient ways to approximate spectral quantities such as the density of states (DoS). In this paper, we numerically investigate two distinct approaches for approximating the DoS of incommensurate Hamiltonians for small values of the incommensurability parameters $\epsilon$ (e.g., small twist angle, or small lattice mismatch): the first employs a momentum-space decomposition, and the second exploits a semiclassical expansion with respect to $\epsilon$. In particular, we compare these two methods using a 1D toy model. We check their consistency by comparing the asymptotic expansion terms of the DoS, and it is shown that, for full DoS, the two methods exhibit good agreement in the small $\epsilon$ limit, while discrepancies arise for less small $\epsilon$, which indicates the importance of higher-order corrections in the semiclassical method for such regimes. We find these discrepancies to be caused by oscillations in the DoS at the semiclassical analogues of Van Hove singularities, which can be explained qualitatively, and quantitatively for $\epsilon$ small enough, by a semiclassical approach.

We introduce a metric-dependent geometric variant of factorization homology in conformally flat Riemannian geometry for $d \geq 2$. Its coefficients are symmetric monoidal functors from a disk category in conformal Riemannian geometry to the ind-category of Hilbert spaces, which we call conformally flat $d$-disk algebras. We prove that their left Kan extensions define symmetric monoidal invariants of conformally flat manifolds. Under suitable positivity and continuity assumptions, the value on the standard sphere recovers the sphere partition function of the associated conformal field theory. For $d\geq 3$, we construct explicit examples using unitary representations of $\mathrm{SO}^+(d,1)$ and harmonic analysis, and show that their operadic structures do not extend to bounded operations on the natural Hilbert space completions.

A classical Davies generator provides a Lindbladian for which the Gibbs state is stationary. Its construction involves precise knowledge of the Bohr spectrum or equivalently state evolution for all times. Recently Chen, Kastoryano and Gilyén proposed a construction involving localisation in time and carried out in the case of finite dimensional Hilbert spaces. The resulting generators are called quantum Gibbs samplers as the corresponding Lindblad evolution is expected to settle to the Gibbs state. In this paper, we show that the localised Davies construction also works for natural classes of unbounded operators, including pseudodifferential operators used in the study of classical/quantum correspondence in Lindblad evolution. Our emphasis is microlocal: we prove that the localised jump operators are themselves pseudodifferential, and hence pseudolocal. The proof involves a novel version of Egorov's theorem valid for all times.

We consider the problem on the existence of two dimensional superintegrable systems in the presence of a magnetic field in the two dimensional Euclidean space. We assume the existence of two integrals of motion, besides the Hamiltonian, that are quadratic polynomials in the momenta. This problem was already studied in the cases where one integral is of Cartesian or polar type [J. Bérubé, and P. Winternitz, J. Math. Phys., 45(5): 1959-1973, 2004]. We continue the investigation by assuming that one of the integrals is of parabolic type and the second integral is of elliptic or (''non-standard'') parabolic type, confirming so far that, on the Euclidean plane, the only two dimensional superintegrable system with quadratic integrals is the one with constant magnetic field and constant electrostatic potential.

In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds admitting a dominated splitting $E^{cs} \oplus E^{cu}$. First, we prove that the smallest Lyapunov exponent along $E^{cu}$, computed with respect to the Lebesgue measure, is computable using observable measures. Then we show that if the Lyapunov exponents along $E^{cu}$ are positive Lebesgue almost everywhere and $E^{cu}$ admits a finest 1-dominated splitting on the support of an ergodic observable measure then $f$ is non-uniformly expanding along $E^{cu}$. As a byproduct, every $C^{1+\alpha}$ diffeomorphism exhibiting a dominated splitting $E^{s} \oplus E^{cu}$ where $E^{cu}$ fulfills the previous assumptions admits an SRB measure.

Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By twisting the post-Hopf product, we provide a combinatorial antipode formula for the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed inverse formula for the Guin-Oudom isomorphism in the context of post-Lie algebras. Especially as a byproduct, we derive a cancellation-free antipode formula for the Grossman-Larson Hopf algebra of ordered trees through a concrete tree-grafting expression.

Making use of the simple fact that all separable complex Hilbert spaces of given dimension are isomorphic, we show that there are just six basic ways to define generalized coordinate operators in Quantum Mechanics. In each case a canonically conjugate generalized momentum operator can be defined, but it may not be self-adjoint. Even in those cases we show there is always either a self-adjoint extension of the operator or a Neumark extension of the Hilbert space that produces a self-adjoint momentum operator. In one of the six cases both extensions work, thus leading to seven basic pairs of coordinate and momentum operators. We also show why there are more ways of defining basic coordinate and momentum measurements. A special role is reserved for measurements that simultaneously measure both.

We study loop ensembles on locally finite rooted trees. The loops are induced by a Poisson process of links on the edges; the transposition-only case is the random interchange process, and the same framework covers loop representations of quantum spin systems. The set of edges carrying at least one untyped link forms i.i.d. Bernoulli bond percolation with retention probability $1 - \exp(-\beta)$, so an infinite loop can occur only if the corresponding link cluster is infinite. Our main results for Galton-Watson trees show both phenomena: if the offspring distribution has finite mean m in $(1,\infty)$, then, conditioned on survival, the loop threshold is strictly larger than the link threshold; in contrast, in the random interchange case a heavy-tail condition on the offspring distribution forces the loop and link thresholds, averaged over both the tree and the link configuration, to coincide at zero. The separation theorem follows from a stronger deterministic criterion based on local loop configurations that cut descendant subtrees out of link clusters.

Recently, several spin chain models have been discovered that admit solutions in terms of "free fermions in disguise." A graph-theoretical treatment of such models was also established, giving sufficient conditions for free fermionic solvability. These conditions involve a particular property of the so-called frustration graph of the Hamiltonian, namely that it must be claw-free. Additionally, one set of sufficient conditions also requires the absence of so-called even holes. In this paper, we present a model with disguised free fermions where the frustration graph contains both claws and even holes. Special relations between coupling constants ensure that the free fermionic property still holds. Notably, the central elements associated with the even holes can be removed by fixing the gauge, revealing our model to be an integrable deformation within the original algebra of free fermions in disguise. The transfer matrix of this model can be factorized in a special case, thereby proving the conjectured free fermionic nature of a special quantum circuit published recently by two of the present authors.

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the corresponding spectral map from material parameters to eigenpairs may exhibit non-smooth or bifurcating behavior. In this work, we develop a theoretical framework identifying sufficient conditions for such non-smooth spectral behavior in the ITP on general domains. We further specialize our analysis to some radially symmetric geometries, enabling a more precise characterization of bifurcations in the spectrum. Computationally, we formulate the ITP as a parametric, discrete, nonlinear eigenproblem and use a match-based adaptive contour eigensolver to accurately and efficiently track eigenvalue trajectories under parameter variation. Numerical experiments confirm the theoretical predictions and reveal novel non-smooth spectral effects.

In arXiv:2211.09069, significant progress was made in decomposing simple products of fermion correlation functions, and in summing over spin structures of superstring amplitudes in genus two under cyclic constraints. In this manuscript we consider part of the same subject using a framework in which one of the branch points of the genus two curve is fixed at infinity. This framework is a direct generalization of the popular one in the case of genus one. We address some of the issues that remained unresolved in our previous paper arXiv:2209.14633. We show that the spin structures of the simple products of fermion correlation functions with cyclic conditions depend only on the Pe-function values at the half-periods of the genus two surface, for any number of factors in the products. Similar to the genus one case, we can provide basis functions to decompose the product. Consequently, the trilinear relations found in arXiv:2211.09069 can be derived from the known set of differential equations of genus two Pe-functions by setting the variables equal to the half-periods of the non-singular and even spin structures, as is the case for genus one. Based on these considerations, we present a procedure for expressing the results of decomposed formulae in terms of the unique genus two theta function for two, three, and four point cases, and discuss a realistic approach for calculating six point function. A general formula for the expression of the results in terms of the theta function for the product of an arbitrary number of the fermion correlation functions is not yet derivable.

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.

We construct new multiparticle models of $\mathcal{N}=4$ supersymmetric mechanics with spin degrees of freedom by employing nonlinear indecomposable supermultiplets ${\bf (1,4,3){\supset\hspace{-1.1em}+}(4,4,0)}$. These systems are proper deformations of those associated with the standard irreducible $d=1, \mathcal{N}=4$ multiplets. In this way we find a new $\mathcal{N}=4$ supersymmetric generalization of U$(2)$-spin rational Calogero system invariant under $d=1$ superconformal group OSp$(4|2)$. One more deformed model reproduces $\mathcal{N}=4$ supersymmetric U$(2)$-spin hyperbolic Calogero system, up to a shift of the Hamiltonian by some U$(1)$ generators.