Papers

In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras $A$ and $B$, are tomographically complete, purifications of a state on $A$ need not be related by unitaries in $B$. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant $B'$ is strictly larger than $A$. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on $A$ by measurements on $B$ is equivalent to the existence of a state-preserving conditional expectation from $B'$ onto $A$. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from $A$ to $B'$. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion $A\hookrightarrow B'$, a left inverse is precisely a conditional expectation, yielding the characterization above.

Let $\mathcal{T}_{n,m} = \mathcal T_n(M_m(\mathbb{C}))$ denote the operator system of all block Toeplitz matrices $ T = (( T_{i-j}))_{i,j=1}^n$ with entries $T_k \in M_m(\mathbb C) $ % T_{k} = \left[ t^{(k)}_{p-q} \right]_{p,q=1}^m. \[ T = \begin{pmatrix} T_0 & T_{-1} & \cdots & T_{-(n-1)} T_1 & T_0 & \cdots & T_{-(n-2)} \vdots & \vdots & \ddots & \vdots T_{n-1} & T_{n-2}& \cdots & T_0 \end{pmatrix} \in M_{mn}(\mathbb{C}). \] We characterize all pure unital completely positive ({\it{ucp}}) maps from $\mathcal{T}_{n,m}$ to $M_m(\mathbb{C})$. Working through the Stinespring isometry $V = (V_1, \ldots, V_n)^t \colon \mathbb{C}^m \to \mathbb{C}^{mn}$ and the matrix-valued polynomial $Q_V(z) = \sum_{i=1}^n z^{n-i} V_i$, we prove that $\varphi$ is pure if and only if it admits a unique pure \ucp extension to $M_{mn}(\mathbb{C})$ if and only if $Q_V$ has degree $n-1$ with all its roots on the unit circle $\T$. Every such pure $\varphi$ induces a \ucp map $\Phi_{Q_V}$ on $C(\mathbb{T}, M_m(\mathbb{C}))$ given by \[ \Phi_{Q_V}(f) = \int_{\mathbb{T}} Q_V(z)^* f(z) Q_V(z)\, dz. \] Let $\mathcal{Y}_m$ be the compact convex set of all \ucp maps from $C(\mathbb{T}, M_m(\mathbb{C}))$ to $M_m(\mathbb{C})$. Endowing $\mathcal{Y}_m$ with the matricial Monge-Kantorovich metric $\rho$, via an analysis of the extreme points of $\mathcal{Y}_m$ together with a point-splitting lemma, we show that the induced maps as above are $\rho$-dense in $\mathcal{Y}_m$. Consequently, if $\Bmn$ denotes the set of normalized $\Phi_{Q_V}$ where $Q_V$ is as above, then the Hausdorff distance $d_H(\Bmn, \mathcal{Y}_m) \to 0$ as $n \to \infty$, extending known results of approximation of positive regular Borel measures on the unit circle to the setting of matrix-valued completely positive maps.

Two alternative approaches for studying Helium-like atomic ions in non-relativistic quantum mechanics are proposed: (I) a numerical approach, based on the Lagrange-mesh method which can easily reach up to 14-15 significant digits in the energy spectrum for any nuclear charge $Z$ with modest CPU time in single processor mode and (II) a highly-accurate, few-parametric interpolation formula for the energies vs. $Z$. The interpolation formula of general nature is proposed, it can be applied to the energies of any excited state of the helium-like sequence. It is based on matching the $1/Z$-expansion at large $Z$ and the Puiseux expansion with integer and half-integer powers around the so-called second critical charge $Z_B$, introduced by F and D Stillinger (1969, 1974), confirmed by the present authors in 2019 for the ground state $1^1 S$, then revisited here, and extended to the excited states in the present work. For example for the first two spin-singlet $1^1 S$, $2^1 S$ and the first two spin-triplet $2^3 S$, $3^3 S$ states this interpolation formula with nine free parameters can reach an accuracy of 10-14 significant digits (s.d.) in the energies for any physically-relevant nuclear charge $Z$, giving absolute accuracy at large $Z$. Many results are obtained for the first time.

Reflection positivity is one of the Osterwalder–Schrader axioms for Euclidean quantum field theory, ensuring that the reconstructed relativistic Hilbert space carries a positive-definite inner product. For free fermionic theories whose covariance is a real rational function of the Dirac operator, we prove that reflection positivity holds if and only if all poles are real, simple, and carry non-negative residues. We also explicitly show the failure of reflection positivity for covariances with exponential regulators.

We prove asymptotic stability of the degree-one vortex in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, for small equivariant perturbations in weighted Sobolev spaces. The abelian Yang-Mills-Higgs model is a classical relativistic field theory on $(1+2)$-dimensional Minkowski space, describing a complex-valued field coupled to an electromagnetic potential and admitting topological solitons known as vortices. This paper is the final and main part of a three-paper series. Under the orthogonal gauge condition used here, perturbations of the vortex are governed by a system of nonlinear Klein-Gordon equations for the dynamical variables, coupled to an elliptic equation for the temporal component of the electromagnetic potential. The linearized operator has continuous spectrum $[1,\infty)$ and a single positive gap eigenvalue (internal mode) of multiplicity two, whose spectral properties, associated distorted Fourier theory, and linear decay estimates are developed in the two companion papers. The main difficulty is the long-time analysis of the coupled radiation–internal-mode dynamics. In two space dimensions the dispersive decay of the Klein-Gordon radiation is relatively weak, while the internal mode decays only on the long time scale dictated by nonlinear radiation damping. At the same time, the Klein-Gordon equations for the radiation contain non-spatially localized variable coefficient quadratic interactions, which cannot be treated perturbatively and require a normal form analysis. We prove decay of the radiation by combining a good-bad decomposition, a flat-sharp decomposition, and a space-time resonance analysis carried out relative to the flat Klein-Gordon flow. The passage between the flat analysis and the Klein-Gordon flow with potential is achieved through ILED and transference estimates derived from the distorted Fourier theory.

We study the linearized dynamics near the degree-one vortex of the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, restricted to equivariant perturbations in the orthogonal gauge. The linearized operator is a selfadjoint matrix Schrödinger operator $\mathbf{M}$ on radial $L^2_{\mathrm{rad}}(\mathbb{R}^2;\mathbb{R}^4)$ with continuous spectrum $[1,\infty)$ and a two-dimensional internal mode at a unique gap eigenvalue $\lambda^2 \in (0,1)$, as established in Part I of our three-paper series on asymptotic stability of the ground state vortex. In this second part of the series, we prove linear estimates for $\mathbf{M}$ for applications in Part III. Specifically, we prove dispersive and local-energy decay estimates, as well as a transference relation which allows us to implement the space-time resonance method with respect to the flat Klein-Gordon operator in the nonlinear analysis in Part III. The engine for proving linear estimates for $\mathbf{M}$ in our approach is the distorted Fourier transform associated with $\mathbf{M}$. The construction of the distorted Fourier transform together with a detailed analysis of the underlying generalized eigenfunctions occupy the first half of this paper. For this we exploit the super-symmetric factorization of $\mathbf{M}$, and the diagonal structure of the super-symmetric partner operator, to relate the problem to the Weyl-Titchmarsh theory of two strongly singular scalar half-line Schrödinger operators.

This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schrödinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schrödinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.

In this paper, we propose an efficient and robust fifth-order finite-volume Hermite weighted essentially non-oscillatory (HWENO) scheme with gradient reconstruction for the compressible Navier–Stokes equations. The key idea is to construct weak-derivative moments for each dissipative variable by taking the arithmetic average of two HWENO interface traces, which are then processed componentwise through the standard scalar nonlinear HWENO reconstruction formula, with only the input moments differing from those used in the convective reconstruction, thereby supplying the gradients needed for the viscous fluxes. This reconstruction achieves fifth-order accuracy using only candidate polynomials of degree at most four, avoids the order reduction that would result from direct differentiation, and introduces negligible additional computational cost. To ensure robustness, especially in extreme test cases involving strong shocks or low densities, we apply a positivity-preserving limiter to the conservative states while retaining the nonlinear HWENO reconstruction for the viscous gradients. This combination is essential for stable computations in flows with strong gradients. Both the conservative and gradient reconstructions share the same compact stencils and candidate-polynomial structure, enabling a single implementation routine without introducing additional algorithmic complexity. Numerical results confirm that the proposed scheme delivers fifth-order accuracy and high resolution, offers competitive computational efficiency, and robustly resolves challenging finite-Reynolds-number Navier–Stokes flows in which the positivity-preserving limiter is actively engaged.

We study the limiting law of local field in the mean field Ghatak-Sherrington model with spin values $\{0,\pm1,\ldots,\pm S\}$ in the high-temperature regime. Using the cavity method and the quantitative overlap and self-overlap estimates of Sheng-Wu~\cite{SW24}, we prove a central limit theorem for the non-centered cavity field. As a result, we then identify the local field limiting law as a random finite mixture of Gaussian distributions. The proof is based on moment method and gives a quantitative error bound.

We develop a diffeological framework for the geometry of Euler–Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler–Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler–Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

Let $\mathcal H$ be a finite-dimensional complex Hilbert space, and let $w(P,\mathsf M)$ be a normalized probability weight assigned to an outcome projection $P$ as it occurs in a projective measurement $\mathsf M$. For fixed $P$, let $G_P\cong\mathcal U(P^\perp)$ be the group of unitaries acting identically on $\operatorname{ran}P$ and arbitrarily on $P^\perp$. We classify the $G_P$-orbits of projective measurements containing $P$: two measurements lie in the same orbit exactly when the multisets of ranks of their complementary outcomes agree. The orbit with profile $\lambda=(r_1,\ldots,r_k)$ is a compact homogeneous space of real dimension $(d-\operatorname{rank}P)^2-\sum_j r_j^2$. On maximal rank-one measurements there is one orbit, so context independence is equivalent to $G_P$-invariance, with equality of the corresponding uniform defects; invariance under two-level complementary unitaries already suffices. For arbitrary projective measurements, every context containing an outcome $P\neq I$ coarsens to the unique binary context $\{P,I-P\}$. Consequently, refinement consistency alone is equivalent to context independence. The finite orbit space carries a natural rank-profile refinement graph, whose diameter is $d-\operatorname{rank}P-1$ when $P\neq I$. We prove a stability theorem that compares the binary-coarsening path with a shortest path in this graph followed by one complementary unitary. In dimension at least three, Gleason's theorem converts the maximal-context invariance condition and the all-context refinement condition, on their respective domains, into the Born form $\operatorname{Tr}(\rho P)$. The results are structural characterizations, not independent physical derivations of context independence.

We develop an integrability framework for Hamiltonian dynamics on con-locally conformal symplectic (COLCS) manifolds, odd-dimensional quadruples $(M,\Omega,\theta,\eta)$ where $d\Omega=\theta\wedge\Omega$, a closed $1$-form $\eta$ determines a codimension-one distribution on which $\Omega$ is non-degenerate, and $R$ is the Reeb vector field satisfying $\iota_R\Omega=\iota_R\theta=0$, $\iota_R\eta=1$. This class simultaneously generalises LCS and cosymplectic manifolds and provides a natural arena for time-dependent Hamiltonian systems with twisted differential $d^\theta=d-\theta\wedge$. The COLCS bracket is introduced on $C^\infty(M)$ and shown to be a Lie bracket that induces Poisson structures on the subalgebras of $\theta$-strong ($\theta(X_H)=0$) and $R$-strong ($R(H)=0$) functions. A Lie-type integrability theorem is then established: given $2n-k$ functionally independent first integrals with $k$ of them $\theta$-strong and generating a solvable Lie algebra under the COLCS bracket, the flow is integrable by quadratures on the common level set. Finally, scaling symmetries of degree $(\Lambda,\beta,\gamma)$, defined by $L_X\Omega=\beta\Omega$, $L_XH=\Lambda H$, $L_X\eta=\gamma\eta$, are studied: they rescale $X_H$ by $(\Lambda-\beta)$, generate families of first integrals, and imply structural primitives for $\Omega$ and $\eta$.

For any finite modular tensor category A, we show that the associated derived $\infty$-category D(A) supports a topological quantum field theory in dimension 3. This TQFT takes the form of a symmetric monoidal functor from an $\infty$-category of surfaces with markings by objects in D(A), and appropriately decorated bordisms, to the $\infty$-category of dg vector spaces. We show that the state spaces in this theory are naturally identified with linearized mapping spaces for D(A). Though our TQFT requires markings from the derived $\infty$-category, we show that all markings can be removed after taking a homotopy truncation. The resulting unmarked TQFT produces projective mapping class group actions on cohomology in dimension 2, and in particular a projective SL_2(Z)-action on Hochschild cohomology. We expect these mapping class group actions to recover those of Lentner et al. arxiv:2003.06527 and Schweigert-Woike arxiv:2004.14343. In dimension 3 we obtain power-series valued knot invariants, and power-series valued invariants for modular tensor categories. We also obtain power-series invariants for closed 3-manifold, though these can already be calculated at the abelian level. Our derived field theories are proposed as mathematical formalizations for topological A-twists of certain N=4 supersymmetric QFTs, in dimension 3. Following physical principles, we discuss the possibility of deforming our TQFTs along local systems via an analogous (conjectural) deformation of quantum group representations along the Langlands dual group.

We study the recursion coefficients of orthogonal polynomials and their associated Krylov dynamics in random matrix models with high-degree and possibly asymmetric polynomial potentials. We develop a moment recursion method that, when combined with the recursive algorithm, provides an efficient construction of the recursion coefficients. We also obtain their large-$n$ asymptotic behavior for general asymmetric potentials; for $Nw_d=1$, the leading asymptotic form of $R_n$ reproduces Freud's conjecture. We apply this framework to an asymmetric quartic potential and to the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In both models, the recursion functions capture the overall qualitative behavior of the recursion coefficients, and the gradient catastrophes of the recursion functions are associated with “chaotic” transition regions in the recursion coefficients. For the quartic potential, such regions can occur in both $R_n$ and $S_n$, whereas the DSSYK model can exhibit multiple transition regions in $R_n$, with the recursion function remaining accurate in the smooth intervals between them. Finally, we compute the corresponding spread complexity and find that transition regions do not qualitatively modify its behavior, while a two branch structure produces early time oscillations followed by monotonic growth.

We establish a global, worst-case non-identifiability theorem for periodically driven finite-dimensional quantum systems. On the unrestricted smooth periodic Hamiltonian class, we show that the period-averaged Fubini-Study metric component of a fixed initial state cannot, in general, be reconstructed from exact one-period propagators indexed by every starting time and external parameter. Thus, even complete starting-time-resolved endpoint data are insufficient to determine this intra-period geometric quantity. The obstruction is characterized exactly. The starting-time-indexed endpoint data determine the conjugation path of the monodromy, but not its particular unitary lift. On each fixed-monodromy slice, the observational fibres are precisely the right orbits generated by smooth parameter-dependent based loops taking values in the pointwise centralizer of the monodromy. The Fubini-Study functional is not invariant under this fibre action and therefore does not factor through the endpoint observation map. An explicit real-analytic two-level witness demonstrates the obstruction within a commuting, one-generator Hamiltonian family, so neither non-Abelian time ordering nor Floquet-logarithm ambiguity is required. A continuous family of Hamiltonians produces identical starting-time-indexed one-period endpoint data while yielding different, and on the unrestricted class arbitrarily separated, period-averaged Fubini-Study geometry. Non-identifiability persists under any prescribed uniform bound on the parameter derivative of the Hamiltonian. The result is a deterministic global statement, not a claim of generic non-identifiability or experimental impossibility. It identifies a precise information gap between complete one-period endpoint data and full intra-period dynamics: the missing information is the unitary lift of the observed monodromy path rather than ordinary scalar phase freedom.

We investigate a class of Ricci-flat Kähler metrics on generalized conifolds constructed via gauged linear sigma models (GLSMs) with indefinite signature. By introducing shadow coordinates (superfields) entering the sigma model with negative signature kinetic term, we show that these GLSMs yield explicit Ricci-flat metrics on complex cones over products of projective spaces. We provide a general formula for the resulting Kähler potentials, along with detailed examples. Our results suggest new directions for the study of Calabi-Yau metrics and toric geometry, and raise interesting questions about the geometric meaning of indefinite signature models. We also give an interpretation in terms of a novel generalized Kähler gauging.

Two-dimensional translation-invariant topological CSS codes on qubits are known to be locally equivalent, after coarse-graining, to stacks of toric codes. However, existing constructions generally break more translation symmetry than is required to remove anyon-permuting translations, leaving open whether any further obstruction exists. We prove that no such obstruction occurs: after passing to the maximal anyon-preserving superlattice, every such code admits a local unitary decoupling into toric codes and product states. We further provide an efficient algorithm for explicitly constructing the decoupling map, together with bounds on the required supercell size and operator spreading. The decoupling requires no additional ancillas in generic cases and extends to finite systems with suitable boundary conditions.

Koszul-complex stabilizer models unify the toric-code hierarchy and bivariate-bicycle codes in one homological framework. To study translation-symmetry-enriched topological (SET) phases in these models and, more generally, in translation-invariant Calderbank-Shor-Steane (CSS) codes with stabilizer maps $(\varphi_X,\varphi_Z)$, we introduce the associated topological superselection profile $\mathscr S_\sigma:=\tau_{\geq1}\mathbf R\!\operatorname{Hom}_R(\overline{\operatorname{coker}\varphi_\sigma},R),\ \sigma=X,Z$. Its cohomology layers $E_\sigma^\ell:=H^\ell(\mathscr S_\sigma)\cong_R\operatorname{Ext}_R^\ell(\overline{\operatorname{coker}\varphi_\sigma},R)$ encode sectors, fusion, and translation action. When finite, the $\ell=1,2,\ldots$ layers describe pointlike, looplike, and higher-dimensional excitations; $\mathscr S_\sigma$ retains inter-layer gluing. For regular Koszul models, we compute $E_\sigma^\ell$ explicitly, find $\mathscr S_\sigma$ single-layer, and obtain finite-size $Z$-logicals from $\operatorname{Tor}$. Single-layer means that at most one positive-degree layer $E_\sigma^\ell$ is nonzero. Our matrix-level Schanuel-lemma method proves rigidity: under a finite-resolution hypothesis, the nonzero layer and degree uniquely determine a topological CSS code's translation SET order at the stable translation-invariant Clifford level. For prime qudits the hypothesis is automatic, and a finite layer's translation kernel gives the exact minimal coarse graining to a toric-code stack. We realize admissible single-layer data. Beyond this regime, split and nonsplit extensions of 3D qubit toric codes yield eight models sharing all identical $E_\sigma^\ell$ layers with trivial translation action but having distinct size-dependent ground-state degeneracies, hence distinct translation SET phases. Thus inter-layer gluing carries SET data invisible to individual layers.

We study rooted Gibbs-DLR measures in the directed polymer model on $\mathbb{Z}^2$ with an ergodic disorder distribution which satisfies an additional mild hypothesis. We prove that the set of extremal rooted Gibbs-DLR measures is closed and totally ordered, characterize extremality in terms of path coalescence, and show that each fully supported extremal rooted Gibbs measure canonically generates a globally consistent and coalescing family of extremal rooted Gibbs measures indexed by all lattice sites. These families are, moreover, totally ordered. Building on this structure, we prove strong existence and strong uniqueness of the associated Busemann process, together with an $L^1$ continuity theorem for the shift-covariant Busemann cocycles which is joint in the inverse temperature, the tilt parameter, and the random environment. This yields, as a corollary, in-probability continuity of the generated extremal Gibbs measures corresponding to directions of differentiability under bounded i.i.d. perturbations of the weights. In positive temperature, it shows in-probability convergence of the generated extremal Gibbs measures. At zero temperature, it also yields quenched subsequential large deviation principles for the corresponding positive-temperature rooted Gibbs-DLR measures on path space. The rate functions are determined by a zero-temperature Busemann cocycle and vanish precisely on the infinite geodesics generated by that cocycle.

A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.

Higher-order networks, represented as hypergraphs, enable direct modeling of multi-body interactions of arbitrary size. Hypergraph representations of real-world systems have been observed to exhibit high simpliciality — the tendency for subsets of hyperedges to also appear as hyperedges — yet the generative mechanisms responsible for this structure are poorly understood. We introduce a generalized preferential attachment hypergraph model in which both hyperedge size $Y_t$ and the number of new nodes per step $X_t$ are drawn from arbitrary distributions, and derive analytically, using a mean-field approximate master equation approach, that the stationary hyperdegree distribution follows a power law whose exponent depends only on the ratio $p = E[X_t]/E[Y_t]$, independent of the shapes of the underlying distributions. Crucially, both $X_t$ and $Y_t$ can be estimated directly from any timestamped hypergraph dataset via a backward-stepping procedure, enabling the model to be fit without parametric assumptions. Applying a nonlinear extension of the model to eight real-world hypergraph datasets, we find that the simplicial fraction increases monotonically with the strength of preferential attachment up to the gelation transition at $\alpha > 1$, establishing preferential attachment as a simpliciality-enforcing mechanism.

Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $\Gamma$. Let $\Lambda$ be the Dirichlet-to-Neumann map on $\Gamma$ and let ${\rm det}_\zeta(\Lambda)$ be its (modified, i. e. with zero mode excluded) $\zeta$-regularized determinant. It is well-known that the quantity ${\rm det}_\zeta(\Lambda)/|\Gamma|$ (where $|\Gamma|$ is the length of $\Gamma$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillopé \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillopé using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.

Diffusion-based quantum state tomography (QST) has shown promising results, but all existing methods implicitly adopt a single parameterization (typically Cholesky) without systematic evaluation. We present the first design space study of density matrix parameterizations for diffusion QST, introducing a geometric framework based on the Jacobian Gram matrix $\mathbf{J}^\top\mathbf{J}$. Our calibration of seven parameterizations at 2- and 3-qubit scales, validated by end-to-end training, reveals that geometric conditioning alone does not predict end-to-end performance: at 3-qubit scale, Hermitian direct ($\kappa = 2.0\times$) performs worse than Cholesky ($\kappa = 27\times$) at all shot levels—a $13.5\times$ isotropy advantage that translates into a fidelity disadvantage of up to $+0.51$. The 2-qubit ranking (Hermitian $>$ Bloch) reverses at 3 qubits (Bloch 0.907 vs.\ Hermitian 0.394). We provide a geometric explanation: unbounded parameterizations suffer projection-induced information loss because the PSD constraint couples diagonal and off-diagonal coordinates in ways the unconstrained model cannot respect, whereas the Bloch representation places the maximally mixed state at the center of the valid region, minimizing projection loss.

Let $\psi({\mathbf x})$, ${\mathbf x} \in{\mathbb R}^{3N}$, be an eigenfunction of the $N$-particle atomic Schrödinger operator. We consider the one-particle density matrix $\gamma(x, y)$ and one-particle kinetic energy density $\varkappa(x, y)$, $x, y\in {\mathbb R}^3$, associated with the eigenfunction $\psi$. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators ${\sf{\Gamma}}$ and ${\sf{K}}$ with kernels $\gamma(x, y)$ and $\varkappa(x, y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues $\lambda_k({\sf{\Gamma}})>0$ and $\lambda_k({\sf{K}})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf{\Gamma}}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,\lambda_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $\psi$. These asymptotics are determined by the singularities of the function $\psi$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $\psi$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $\psi$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues $\lambda_k(\sf{\Gamma})$ and $\lambda_k(\sf{K})$. The asymptotic formulas take the form \[ \lim_{k\to \infty} k^{\frac{10}{3}} \,\lambda_k({\sf\Gamma}) = \big(A_{asym}\big)^{\frac{10}{3}},\quad \lim_{k\to \infty} k^{\frac{8}{3}} \,\lambda_k({\sf K}) = \big(B_{asym}\big)^{\frac{8}{3}}, \] where $A_{asym}$ and $B_{asym}$ are non-negative constants given explicitly in terms of the gradient of $\psi$.

We develop an unfitted ghost finite element method for elliptic interface problems with discontinuous diffusion coefficients and apply it to the electrostatic simulation of low-dimensional semiconductor devices. The proposed approach is based on a fixed Cartesian grid and represents the geometry by level-set functions, avoiding mesh generation and remeshing even in the presence of interfaces. A snapping-back-to-grid strategy is used to control the small-cut-cell issue, while interface conditions are weakly enforced by a symmetric Nitsche formulation. The method is first validated on benchmark elliptic interface problems with different geometries and coefficient jumps, showing second-order accuracy for the solution and first-order accuracy for its gradient. As an application, we consider a graphene field-effect transistor described by a self-consistent drift-diffusion-Poisson model. The electrostatic potential is computed in a two-dimensional oxide/graphene/oxide structure, while charge transport in the graphene layer is modeled by one-dimensional bipolar drift-diffusion equations in the degenerate case and by including a field-dependent mobility model. The coupled nonlinear system is solved by a damped fixed-point iteration combined with a domain-decomposition treatment of the three-layer geometry. Numerical simulations reproduce transfer characteristics with a clear transition from an OFF state to an ON state and show the influence of the gate voltage on the two-dimensional electrostatic potential. The results also indicate that the effective thickness and discretization of the graphene layer affect the transverse potential profile, supporting the use of a full two-dimensional electrostatic description with explicit oxide/graphene interface conditions.

We present an analysis on the geometrical and physical nature of the $p_x + ip_y$ superconductor on the square lattice, with an emphasis on the topological phase transition at half-filling. We develop a local topological marker from specific Dirac points within the Brillouin zone, which is introduced via the addition of two one-dimensional (1D) $\mathbb{Z}$ $(\mathbb{Z}_2$) invariants defined on the Bloch sphere. We relate this topological marker to the electron spectral function integrated on frequency through the local momentum-resolved electron density, which may be measured via Angle Resolved Photoemission Spectroscopy (ARPES), and show that it remains well-protected including temperature effects. Integrating on a small area around a specific point in momentum space associated to the measure uncertainty, this also reveals the Van Hove logarithmic profile of the density of states in the derivative of the local marker while preserving the topological information. Topological transitions correspond to a protected semi-metal. We analyse the real space representation of this topological marker from correlation functions. We present physical responses such as the topological superfluid density.

We develop a classical model-theoretic representation of partial Boolean algebras and use it to formulate quantum-like uncertainty without abandoning classical propositional logic. Given a surjection from the initial formulæ\ of a propositional language onto a partial Boolean algebra $(\mathcal V,\Pi)$, we construct a consistent theory $\mathcal T_g$ whose core—the ordered set of equivalence classes of initial formulæ—is isomorphic to $(\mathcal V,\Pi)$. Its models are characterized as upward-closed clusters, yielding a model-theoretic formulation of KS-colourability: an $n$-dimensional partial Boolean algebra is KS-colourable exactly when the induced theory has a model meeting every pre-frame in one primitive formula. For finite-spectrum observables, uncertainty is defined by the number of locally admissible atomic outcomes. Dispersion-free models are characterized by the singleton pre-frame condition. With an additional measurement-update postulate, a finite example shows how measurement of an incompatible observable can destroy sharpness, providing a model-theoretic form of back-action. A $12$-vertex partial Boolean algebra is KS-colourable, whereas a rigorously constructed $140$-vertex, four-dimensional partial Boolean algebra is not, as shown by a parity argument; its incidence structure is isomorphic to the Peres $24$-ray, $24$-basis configuration in $\mathbb R^4$. Macroscopic interpretations show that these phenomena arise from the organization of propositions and models rather than from nonclassical deduction. Finally, we relate certain models to the probability-one propositions of pure states and density operators, while emphasizing that such certainty models do not determine the full quantum state.

The motion of a projectile released from a simple pendulum is analyzed, with particular emphasis on investigating the optimal release angle that maximizes the horizontal range and the corresponding maximum range. This system serves as a simplified model of the Tarzan jump problem. Using simple analytical methods, the optimal release angle is shown to be characterized by a cubic equation and to increase with the initial velocity. In addition, asymptotic expressions for both the optimal angle and maximum range are derived in the limits of low and high initial velocity.

We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang–Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang–Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang–Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang–Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework.

Bernoulli-type waveforms for modified Emden nonlinear oscillators of arbitrary natural power $q$ are obtained through a generalized commutative factorization approach. These oscillators display a well-defined odd-even dynamical dichotomy, which is discussed in detail: the odd-$q$ cases entail isochronous oscillators whose period $T = 2\pi/\omega$ is independent of amplitude and initial conditions, while the even-$q$ cases display underdamped behavior. The Lagrangian formulation is presented in the Lurie's dissipative description. The isochronous regime and the period of the solutions in the odd case are also confirmed through a generalized polar-coordinate analysis in the spirit of Sabatini's work. The absence of periodic orbits for even $q$ is shown to be a consequence of the Bendixson-Dulac criterion applied to the radial velocity function. Explicit waveforms and their phase portraits are presented for $q = 1, 2, 3, 4$, along with the non exponential envelope formulas for the damped cases and singular-region bounds for the isochronous ones. A few possible applications are also mentioned.