Papers

Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.

Quiver-invariant dualities relate distinct 4d N=1 supersymmetric gauge theories that arise as worldvolume theories on a D3-brane probing the same toric Calabi-Yau 3-fold. These dual theories share an identical quiver and differ only in their superpotentials. We show that quiver-invariant dualities are realized by tilting mutations on the brane tilings that realize these theories. Among the 42 toric phases of the H1121 model, we identify three general families of tilting mutations, all characterized by the reversal of the orientations of zig-zag paths within the mutation region of the brane tiling. We present new examples of quiver-invariant dualities, given by five doublets and one triplet of toric phases with identical quivers, for which we verify that the brane tilings related by the tilting mutations correspond to the same toric Calabi-Yau 3-fold H1121.

The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the global topology of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws $b_n\sim n^{1/\beta}$ for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as $n^{-1/3}$ at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate $\alpha = \pi T$, it can be extracted from the subleading offset $b_0 = \pi T(\Delta - \frac{1}{2})$. These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.

The quantum-mechanical two-sided Bogoliubov inequality provides upper and lower bounds for the free energy required to separate a system of interacting particles into independent subsystems. The bounds can be calculated straightforwardly from the ensemble average of the interface energy, bypassing the direct evaluation of the free energy. In this work, we generalize the two-sided Bogoliubov inequality to arbitrary von Neumann algebras by employing the Araki-Uhlmann relative entropy and the framework of unbounded perturbation theory of KMS states. Furthermore, we obtain variational expressions for the relative free energy that extend existing bounded-perturbation principles to the unbounded setting. Crucially, these mathematical developments yield a physically well-founded thermodynamic criterion for the quantification of entanglement in infinite-dimensional systems.

One hallmark of exactly solvable KPZ random growth models is product-form invariant measures. In the setting of exponential last-passage percolation (LPP), this corresponds to the independence of Busemann increments along any down-right path. However, this independence breaks down when multiple asymptotic directions are considered simultaneously, owing to the fact that jointly invariant measures are not jointly product-form. This paper shows that the failure of independence is one-sided: Busemann increments across arbitrary directions are negatively associated. As an application, we derive an exponential concentration inequality for sums of Busemann increments on the diffusive scale, even when the increments are not independent. While our argument relies on a Burke property that is special to exponential weights, all other proof ingredients$\unicode{x2014}$including hidden LPP monotonicities and braid relations for queueing maps$\unicode{x2014}$hold for arbitrary weights.

Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in $2$- and $3$-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects $\mathcal{T}_\mathcal{Z}$ associated to a reflection defect TQFT carries the natural structure of an $O(2)$-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, $\mathcal{T}_\mathcal{Z}$ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).

An algorithm is presented for Kraus decomposition of a completely positive operator over separable (countably-infinite-dimensional) Hilbert spaces, together with an elementary proof that the generated sum convergences in strong-operator topology. This improves on the standard, nonconstructive, proof by fusing the abstract problem with practical process tomography. Kraus operators are generated one-by-one, each having one more guaranteed zero matrix entry than the previous one. In this way, the stream of outputs of the algorithm provides a coherent family of Kraus decompositions of restrictions of the target CP map to ever-larger subspaces.

This article investigates the effective mass density for acoustic and elastic wave propagation in layered media, with emphasis on its transition between these regimes. Conventionally, it is possible to recover the governing equations of acoustics from those of elasticity via the no-shear limit. However, when considering an effective medium, the anisotropic effective mass density typically obtained for acoustics differs from the isotropic one found in elasticity. Furthermore, direct investigation of this limit is hindered by the fact that the effective mass density is independent of the shear modulus $\mu$. To tackle this problem, a transfer matrix approach is combined with Bloch's theorem to derive the effective wavenumber for a periodic material whose unit cell consists of two layers with different densities. The effective mass density is obtained analytically from the wavenumber, allowing for its evaluation in both regimes. Moreover, for the first time, a description of the transition from isotropic elastic to anisotropic acoustic effective density in a periodic layered medium is provided. Additionally, the existence of band structure in $\mu$-space is demonstrated, along with exceptional points, where compressional and shear modes coalesce and waves become evanescent, which heavily affects the behaviour of the effective density in the elasto-acoustic transition.

We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.

Stabilizer Rényi entropy quantifies the nonstabilizerness of a quantum state through Rényi moments of its Pauli expectation-value distribution. Its standard form sums over Pauli-string degrees and retains only the total Rényi weight. We introduce a fugacity-resolved partition function for the critical transverse-field Ising chain that resolves this degree in the balanced Majorana representation and generates its full counting statistics. This Ising problem extends beyond a single model: exact decimation identities and the correspondence with the half-filled \(XX\) chain establish it as a common finite-size building block for stabilizer and computational-basis Shannon–Rényi entropies. We map the fugacity-resolved all-minors sum, for every positive Rényi index, to a checkerboard-weighted discrete Selberg ensemble on a half-filled doubled root lattice. For positive integer indices, it reduces to a finite aliased Dyson constant term. At \(\alpha=\tfrac12,1,2\), determinant and Pfaffian compressions yield product formulas. At \(\alpha=4\), the generic-fugacity problem admits an exact inverse Jack–Kostka representation, while at unit fugacity a complementary middle-minor identity relates it to the square of the \(\alpha=2\) result. These generating functions determine the balanced-degree statistics. Complement symmetry makes the distribution symmetric about \(k=L/2\), with an index-dependent width. It is exactly binomial at \(\alpha=1\), has variance proportional to \(L\) at \(\alpha=\tfrac12\), and develops an \(L\log L\) enhancement at \(\alpha=2\). After variance rescaling, the centered distributions converge to Gaussian limits at all three indices. Beyond these solvable cases, finite-size numerics reveal an evolution from a central peak to a symmetric bimodal profile and eventually to endpoint dominance as the Rényi index increases.

We present the formulation of a boundary value problem that models the coupled behavior of a three-dimensional diffusive solid with one-dimensional diffusive fibers embedded inside it. We introduce a variational statement of the problem that identifies the linked diffusive fields as energy minimizers under a coupling constraint. This saddle-point problem is proved to be well posed. Then, we introduce a finite element discretization of the proposed boundary value problem, and we prove the convergence of the finite element solution to the exact one. The most significant feature of this approximation is that the meshes of the bodies need not be conforming. Numerical examples confirm the theoretical results

Let $\mathrm{CHL}_N$ be the cylindrical Hastings–Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $\lambda>0$, and let $\omega_{N,\lambda}$ be its tree-completion time — the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\mathbb{E}[\omega_{N,\lambda}]\le(1+\varepsilon)(\log N)/(2\lambda)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[\omega_{N,\lambda}]}{\log N}=\frac{1}{2\lambda} \qquad\text{for every fixed }\lambda>0 . \]

This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $\lambda$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.

A finite group-valued temporal reference frame defines a correlated random-unitary process rather than a list of independent channels. We identify a regime in which its complete trajectory law is an exact operational coordinate. If the carrier contains every irreducible representation of the finite group $G$, one ancilla-assisted input resolves all group elements, and for arbitrary laws $\mu,\nu$ on $G^n$ the strategy half-distance equals $d_{\mathrm{TV}}(\mu,\nu)$. Optimal output-side causal post-processing likewise reduces to classical convolution on $G^n$. This converts temporal hiding into a $q$-ary coding problem. Uniform coset laws of an $[n,k]_q$ code are identical on every set of fewer than $d(C^\perp)$ slots and perfectly distinguishable globally. For $M$ sectors with leakage at most $\eta$ from $t$ selected slots and global decoding error $\epsilon$, we obtain $(1-\epsilon)\log M \leq (n-t)\log q+\eta+h_2(\epsilon)$. For nested codes $C\subset D$, the payload $R=\dim D-\dim C$, invisible depth $t=d_{\mathrm{rel}}(C^\perp,D^\perp)-1$, and sector distance $d_{\mathrm{rel}}(D,C)$ obey $R+t+d_{\mathrm{rel}}(D,C)-1\leq n$ and $R+t+2e+f\leq n$, where $e$ and $f$ are adversarial errors and known erasures. Nested generalized Reed-Solomon codes attain the bounds in their existence range. We also give the exact projection-rank leakage profile and show that hiding arbitrary coherent superpositions from $t$ slots is precisely quantum erasure correction, yielding $\log_q K+2t\leq n$. The results separate classical trajectory privacy from coherent temporal privacy and quantify the robust payload hidden from reduced process tomography.

Ice I admits cubic, hexagonal, and mixed layer stackings, but rigorous comparisons of its proton configurational entropy have focused on the two ideal endmembers. We identify every cyclic uniform-registry ice-I stacking with a word in a nonnegative layer-transfer operator and its transpose. Applying the Schatten-Hölder inequality to every even-length word proves that alternating hexagonal stacking maximizes the ice-rule configuration count at each common finite cross-section. Consequently, hexagonal ice maximizes the thermodynamic configuration constant among all periodic uniform-registry ice-I polytypes. We also construct rigorous lower and upper bounds without finite-size extrapolation. Restricting Nagle's positive even-subgraph expansion to disjoint blocks gives the lower bound; Finner's hypergraph Hölder inequality and rational Collatz-Wielandt certificates for two-replica prism operators give the upper bounds. The resulting configuration constant $w$, whose logarithm is the entropy per molecule in units of Boltzmann's constant, satisfies $1.503360 \le w \le 1.540196$, with the sharper cubic-ice ceiling $w \le 1.527699$. The same bracket controls lower and upper configuration growth rates of fixed aperiodic stackings along the specified prism-tileable exhaustions. These results establish hexagonal stacking as a possibly nonunique maximizer; equality of the cubic and hexagonal entropy constants remains unresolved.

We construct a nonlinear electrodynamics (NLED) model for a static magnetic black hole. Rather than imposing a regular center, we require a Maxwell weak-field limit, finite magnetic self-energy, a positive and subluminal electromagnetic cone, the standard energy conditions, and a static exterior that satisfies known sufficient stability criteria. A positive mixture of power kernels restricts the exponent to $1/4<\gamma\leq1/2$. For the minimal two-kernel model, with $\gamma_1=1/3$ and $\gamma_2=1/2$, the field equations admit an exact solution in terms of incomplete beta functions. For the branches with nonnegative Schwarzschild mass parameter $M_0$, the metric function is strictly increasing, so there is at most one positive-radius horizon and no inner Cauchy horizon. We construct the maximal extension and show that the black-hole, horizonless, and critical branches have spacelike, timelike, and null singularities, respectively. The representative black-hole families have positive temperature and negative fixed-charge heat capacity. Electromagnetic waves split into ordinary and extraordinary optical branches. We calculate their photon spheres, shadow radii, instability rates, and illustrative Event Horizon Telescope size bands. We also construct ISCO-truncated thin-disk images. The full images remain close, but jointly normalized residual maps and radial profiles reveal a coherent branch-dependent shift near the lensed inner edge and the critical region. The extraordinary branch moves the critical curve outward and partly compensates the shadow reduction caused by magnetic charge.

We propose a Multi-Resonant Double-barrier Quantum-dot-in-a-Well (MR-DWELL) dosimeter with a tunable trade-off between sensitivity and radiation tolerance, realized through graphene-assisted electrostatic screening. The sensing mechanism relies on radiation-induced trapped charge that shifts and broadens resonant tunneling states in the DWELL heterostructure. The~dimensionless capacitance ratio $\eta=C_Q/C_{\mathrm{geo}}$ is introduced as the universal design parameter: it~controls the screening factor $S=\eta/(1+\eta)$ and thereby the trade-off between sensitivity and radiation hardness. For $\eta=2.09$ the effective low-dose shift coefficient is reduced threefold (from $0.14$~to $0.045~\mathrm{meV/mGy}$), while the screened architecture retains $\sim\!83\%$ of the peak tunneling current and a peak-to-valley ratio $\sim\!10$ after $1~\mathrm{MGy}$, thereby preserving the dosimetric sensitivity required for practical readout. The~second-derivative spectroscopy $d^2I/dV^2$ is identified as a robust experimental readout. A~reduced-order Landauer model calibrated against literature data is employed for the screening physics, while a self-consistent NEGF-Poisson-trap framework is formulated for future quantitative validation. Explicit design rules for three operating regimes (high sensitivity, balanced, maximum hardness) are provided, together with a three-population trap model that conceptually enables dose-rate discrimination in the FLASH radiotherapy regime. A~comparative analysis of model vintages and their methodological foundations is included to guide interpretation of numerical predictions.

The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and $\beta$-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier–Stokes–$\beta$ realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved–reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson–Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the $k^{-5/3}$ inverse-energy and $k^{-3}$ forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.

The quantum lattice Boltzmann (QLB) scheme of Succi and Dellar advances a four-component Dirac spinor on a lattice by a fixed sequence of local, exactly norm-preserving operations: a basis rotation, a collision, a streaming shift, and the inverse rotation. This unitarity is a structural property of the scheme, not an approximation, which suggests that a QLB time step should map onto a sequence of quantum gates. Here we make that mapping explicit. We give a gate-level construction of every operation of the three-dimensional Dirac QLB scheme: the fixed rotation gates, the collision gate, the streaming shift as a controlled increment on a position register, the position-dependent potential as a phase oracle, and periodic and reflecting (bounce-back) boundary conditions as unitary circuits. We then compose them into single-axis, two- and three-dimensional time steps. On a state-vector emulator the resulting circuits reproduce the classical QLB solver to machine precision (maximum density deviation between $3.7\times10^{-12}$ and $1.0\times10^{-17}$ across the one-, two-, and three-dimensional tests), so the circuits are the scheme rather than an approximation of it. The scope is narrow: we establish that the Succi-Dellar theory can be implemented on a (gate-model) quantum computer, and report the associated gate counts. We make no claim of computational advantage; state preparation, measurement, and asymptotic cost are discussed as open questions. All operators, circuits, tests, and figures are reproducible from the open-source quantumKineticMethods library.

In previous work, we identified, for every $p\geq3$, a unique first critical temperature $\beta_1^p$ and proved that the Parisi measure is replica symmetric (RS) for $0<\beta\leq\beta_1^p$ and one-step replica symmetry breaking (1-RSB) on a nonempty interval immediately above $\beta_1^p$. In this sequel, we determine the rest of the phase diagram. We prove that there is a unique second critical temperature $\beta_2^p>\beta_1^p$ such that the Parisi measure is 1-RSB for $\beta_1^p<\beta\leq\beta_2^p$, while for $\beta>\beta_2^p$, $\operatorname{supp}\mu_\beta=\{0\}\cup[q_\beta,q'_\beta],$ for some $0<q_\beta<q'_\beta<1$, and is therefore full replica symmetry breaking (FRSB). Combined with our earlier results, this establishes the two transitions predicted by Gardner for the Ising pure $p$-spin glass.

We present an exact analytical model of axisymmetric gravitational collapse leading to the formation of the Kerr black hole. The model depends only on the total mass ${\cal M}$ and a time dependent rotation parameter $a(v)$, and incorporates both the extremal Kerr black hole and a novel quasi extremal regime without requiring $a\approx{\cal M}$. The evolution develops curvature singularities induced by $a(v)$, which are expected to remain enclosed within a trapped region close to the evolving Kerr radius $r=h(v)$. It also predicts a transient anisotropic exterior curvature decaying as $1/r^2$, which may imprint observable signatures associated with the formation of a rotating black hole. The present construction should therefore be interpreted as an exact analytical description of the final stage of rotational gravitational collapse, immediately preceding the formation of the Kerr black hole.

In this article, we explore the integrable aspects of generalized non-linear fluid equation possessing mixed spatial and time evolutions. This non-linear model has been acknowledged as integrable tool for the non-linear dynamical character- ization of dense material such as in optical fibre, metallic ionic fluid. This model widely has been applied in physical interpretations of electromagnetic propaga- tion through perturbed charged plasma. Here, we use few integrable approaches and attain a broad range of solitary solutions involve deep analytical analysis. These results are furnished with computational simulations, which enrich under- standing about the accurate pictorial flow of energy. We also present a bifurcation analysis for canonical coordinates to examine the impact of parameter variations on non-linear behavior of the systems under consideration. Subsequently, we per- form stability analysis to evaluate the robustness of optical solitons features in the vicinity of equilibrium points. These results establish the reliable and accurate framework for analyzing non linear response of materials for energy propagation by varying physical parameters associated with this non-linear equation model.

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.

The object of investigation in this paper are the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this setup. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet-Neumann (DN) operators, which are introduced for each of the layers. Then, the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulae for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.

Starting from the Standard Model Lie algebra $\mathfrak{g}_{\mathrm{SM}} = \mathfrak{sl}_3 \oplus \mathfrak{sl}_2 \oplus \mathbb{C}$ sitting inside the complex Lie algebra $\mathfrak{e}_7$, we show how to decompose $\mathfrak{e}_7$ into the direct sum of a Lie subalgebra containing $\mathfrak{g}_{\mathrm{SM}}$ and three 32-dimensional subspaces, each of which forms the same representation of $\mathfrak{g}_{\mathrm{SM}}$ as one generation of fermions and their antiparticles. The setting is due to Nasmith, and much of the mathematics is that underlying the $\mathrm{E}_7$ generation unification of Kugo and Yanagida. New features include the derivation, in which as many results as possible rely only on the embedding $\mathfrak{g}_{\mathrm{SM}} \subset \mathfrak{e}_7$, and also the description of each 32-dimensional subspace as a copy of the exterior algebra $\Lambda\mathbb{C}^5$.

The physics of spreading in static networks is well understood through mappings to percolation. We show that spreading dynamics on temporal networks can analogously be mapped to reachability in temporal event graphs. This provides a theoretical and computational framework for a class of processes, such as variants of the susceptible-infected-susceptible model. Without explicit simulations, through the component analysis of event graphs, we obtain epidemic prevalence and derive epidemic thresholds for temporal networks with arbitrary degree and inter-event time distributions, with significant computational advantages as compared to explicit simulations.

We develop an ODE/IM-based formulation for the thermal partition function and the spectral zeta function of the homogeneous Hermitian and PT-symmetric oscillators. For both classes of systems, the quantization condition can be expressed using the counting function $a(E)$, which can be solved via the Destri-de Vega equation of the integrable model. We then express the partition function and spectral zeta function as contour integrals involving the counting function, thereby providing a direct bridge between quantum spectral functions and integrable models.

In this note, we observe that the class of unbounded subnormal operators on quaternionic Hilbert spaces is a solution of quaternionic quantum harmonic q-oscillator whose complex case is addressed by Szefraniec in [19]. We also address the connection of unbounded subnormal operators on quaternionic Hilbert spaces, quaternionic quantum harmonic q-oscillator and quaternionic quantum economics using model theory of unbounded subnormal operators.

We study static, spherically symmetric axion–dilaton solutions of the tree-level four-dimensional string effective action. Using the sigma-model structure of the scalar sector, we show that the full family of static, spherically symmetric and asymptotically flat solutions is obtained as the $SL(2,\mathbb R)$ orbit of the pure-dilaton FJNW solution, and we characterize the associated axion–dilaton charge space. We then analyze the perturbative regime of these solutions by comparing the onset of $\alpha'$ curvature corrections with that of string-loop corrections. Finally, we apply the string–black-hole correspondence criterion to the FJNW family by evaluating the local $\alpha'$ curvature diagnostic at the physical size $R_{\rm typ}$ of a highly excited string state. With the normalization fixed on the Schwarzschild branch, Schwarzschild saturates the nominal $\alpha'$ threshold at the correspondence surface, whereas every scalar-haired FJNW branch lies above it. More generally, independently of the common overall normalization within the matching prescription, every nonzero-hair branch has a larger curvature diagnostic at the correspondence surface than Schwarzschild. Away from the correspondence point, scalar-haired exteriors may remain under $\alpha'$ control at sufficiently weak local self-gravity and may provide perturbatively controlled long-distance fields of highly excited string states whenever the local string coupling also remains weak.

We introduce a second-order mean-field description of the four-dimensional Euclidean $\phi^4$ model within the Wilson–Polchinski renormalization-group framework. The construction is based on the connected amputated Schwinger functions evaluated at the symmetric momentum configurations $ (p,-p,\ldots,p,-p)$, which are decomposed into a momentum-independent component, a component quadratic in $p$, and a higher-order remainder. The first two components are chosen to satisfy a closed nonlinear hierarchy. We prove the existence of solutions to this hierarchy for arbitrary positive bare coupling and establish their convergence to the Gaussian fixed point as the ultraviolet cutoff is removed. In particular, both the momentum-independent and the quadratic momentum sectors are asymptotically trivial.