Papers

We construct a class of global naked singularity solutions for the $(3+1)$-dimensional spherically symmetric Einstein–Maxwell-charged scalar field equations. The solutions arise from $C^{1,\alpha}$ asymptotically flat one-ended characteristic initial data with $0<\alpha\ll1$ and develop a curvature singularity that is not enclosed by an event horizon. The resulting spacetimes are genuinely charged, with a nonzero Maxwell field that satisfies a quantitative positive lower bound up to the singularity and admits an explicit asymptotic profile. As part of the construction, we develop a general framework for constructing naked singularity exteriors, namely, the causal future region of the past light cone emanating from the singularity. The framework accommodates a broad range of singular characteristic data and, in particular, allows for singular behavior of the Maxwell field near the singularity.

The fission matrix (FM) method recasts neutron transport in matrix form, enabling fast, interpolation-based reactor calculations from a pre-computed database of Monte Carlo-derived coefficients. Propagation of the underlying Monte Carlo statistical uncertainty to the FM eigenvalue and eigenvector is rarely rigorous: prior approaches assume independent coefficients without testing this against their actual covariance. This work derives a full-covariance Delta-method framework for propagating uncertainty to keff and the fission source, and evaluates it with two cheaper alternatives, a diagonal-covariance approximation and direct resampling of the FM coefficients, using an OpenMC model of the JSI TRIGA Mark-II reactor. For the all-rods-out case, these three estimators and an empirical benchmark of 100 independent Monte Carlo runs agree within 5.7% for keff and 1% for the fission source. The sign of the diagonal approximation's bias flips between the two statistics levels, so it must be checked case by case. With a global fixed-source FM generation strategy, the diagonal, resampling, and independent-run estimators agree within about 2% and lie roughly 10% below the full-covariance estimate, a gap attributed to covariance sampling noise. The large off-diagonal covariance is nearly symmetric and mostly cancels when propagated, rather than being small or negative as previously assumed. For the FM control-rod interpolation and combination strategy (FM-CRd), keff matches an exact perturbed reference within 60 pcm (about two standard deviations) and the fission source within 0.7%, while the diagonal and resampling estimators agree closely and are conservative by about a factor of two. Given this and the prohibitive memory cost of the full covariance, the diagonal Delta method is recommended as the practical default, with the full covariance reserved for one-time characterization of a system.

We study local eigenvalue statistics and dynamical (de)-localization properties for long-range Anderson models, including the fractional Anderson model. These systems are random Anderson-type perturbations of operators with long-range (non random) hopping terms of the form $|T(n,m)| \sim \|n-m\|^{-(d+2\beta)}$ for $\beta>0$, on the $d$-dimensional lattice. In the presence of a strong enough random potential, these models exhibit dense pure point spectrum with polynomially decaying eigenvectors, almost surely. We show that in this strong disorder regime, and for $\beta>d/2$, the local eigenvalue statistics centered at any $E$ in the deterministic spectrum is a Poisson point process with intensity given by the density of states function $n(E)$. Moreover, we prove that, at strong disorder, there is no dynamical localization for these models, verifying a conjecture of Disertori et al. We achieve this by establishing explicit, sharp lower bounds on the Green's functions fractional moments which imply that large moments of the position operator are infinite. Hence, these models exhibit Poisson eigenvalue statistics and dynamical delocalization, that is, the absence of dynamical localization. In particular, this shows that in dimension $d=1$, the fractional Anderson model, a random perturbation of the fractional Laplacian $(-\Delta)^\alpha$ with $\alpha\in (0,1)$, exhibits Poisson eigenvalue statistics and dynamical delocalization if the disorder is strong and the exponent $\frac{1}{2}<\alpha<1$. This is in stark contrast with what is known for the usual Anderson model that has only nearest-neighbor hopping.

This study develops an energy formulation for linear anisotropic Maxwell–Cole–Cole media with multiple fractional polarization channels, passive conductivity, prescribed currents, and finite-terminal Caputo memory with zero prehistory. A weighted solution class captures the integrable fractional initial layer generated by quiescent polarization. Fractional polarization power then separates into a nonnegative history storage and a nonnegative constitutive dissipation rate. Although the instantaneous rate may diverge at initialization, it remains integrable, while stored history energy and accumulated loss vanish initially. Combined with the Poynting theorem, this decomposition yields a global field-plus-memory identity that establishes electromagnetic passivity, closed-system energy decay, and conditional uniqueness under zero-supply boundary conditions. The framework also recovers the Debye, instantaneous-dielectric, vanishing-susceptibility, and vacuum limits under convergence assumptions. A positive diffusive representation provides a second decomposition whose storage never exceeds the history storage. Their difference is an exact nonnegative weighted variance that vanishes only for a zero polarization history. In agreement with these time-domain results, the finite-time harmonic response approaches the anisotropic multichannel Cole–Cole permittivity, whose Hermitian imaginary part ensures nonnegative cycle-averaged absorption. A reproducible single-channel study shows that fractional order broadens the dielectric-loss band, redistributes wave attenuation, strengthens the integrable initial power layer, and prolongs the step-response tail. An exact electric-step response verifies both energy representations. The formulation connects fractional initial layers, hereditary storage, passivity, diffusive relaxation, dispersion, absorption, and attenuation in a local constitutive model.

The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $\Phi_{A,B,C}$ and $\Phi_{D,E,F}$ on $M_d$, the entrywise bound $(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}$ for all $i,j$ guarantees that $\Phi_{A,B,C}\circ\Phi_{D,E,F}$ is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with $a\ne1$ is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.

The planar loop equations of a Hermitian one-matrix model leave finitely many low moments undetermined. Hankel positivity constrains these moments but does not reveal how a fixed multicut family is embedded in moment space. We introduce a finite-plane diagnostic using the Cauchy kernels $f_n(x)=(z_n-x)^{-1}$ which we use as spectral probes. Their Gram matrix is the positive-semidefinite Pick matrix $P_{mn}$, whose positivity defines a finite-node bootstrap analogous to the Hankel bootstrap. For a fixed regular multicut topology, the nonbranching double zeros of the spectral discriminant, which we call dressed saddles, impose transverse constraints on the corresponding moment-space locus, while its remaining directions are filling fractions. Writing $z=E+i\eta$, $E$ selects a spectral region and $\eta>0$ sets a continuous resolution scale. We use $P(z,z)$ as a finite-plane spectral response and interpret $\nabla_{\mathbf m}P(z,z)$ as a moment-space susceptibility. Near a regular real dressed saddle, this susceptibility is enhanced as $\eta^{-2}$ and aligns with a conormal to the filling-fraction manifold, whereas the leading enhancement cancels along tangent deformations. Responses near several independent saddles can therefore reconstruct the conormal space and, through their common kernel, its tangent space. Combining Pick positivity with reality and analyticity of the resolvent, we obtain alternative analytic derivations of known planar results in quartic and sextic models. The asymmetric quartic makes the one-dimensional filling-fraction geometry explicit, while sextic models exhibit independent conormal directions. These results clarify how local consistency conditions constrain the low moments before global period matching and the equilibrium variational inequality select the equilibrium measure.

[An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on $d$-dimensional tori, $d \le 2$, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If $d=1$, fully periodic symmetric bases exist across all AZC classes. If $d=2$, topological obstructions appear as non-trivial phases characterized by $\mathbb{Z}$ (Chern number) or $\mathbb{Z}_2$ (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.

We develop entropy-stable isothermal-wall conditions for the compressible Navier–Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional $L_2$-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order $p_s+1$ in the discrete $L_2$ norm, where $p_s$ is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.

We justify the vortex-wave system introduced by Marchioro and Pulvirenti in the 1990s as the vanishing-viscosity limit of the two-dimensional Navier-Stokes equations on every prescribed finite time interval. The initial vorticity consists of a point vortex and a smooth, compactly supported background that vanishes near the vortex. The regular component converges strongly to the background vorticity governed by the vortex-wave system, while the concentrated component is approximated by a Lamb-Oseen vortex centered on the limiting point-vortex trajectory. We obtain quantitative estimates for both components, with constants and a viscosity threshold depending on the chosen time interval, thereby providing a global-in-time justification of the vortex-wave system for this class of initial data.

The sudden removal of a potential barrier from a Bose-Einstein condensate (BEC) gives rise to a quantum version of a hydrodynamic dam break and leads to rich wave dynamics that have similarities to other localized defect problems such as domain wall dynamics in spin systems. Denoting $\Delta n$ as the initial difference in density between the upper $n_{1}$ and lower $n_{0}$ reservoirs on either side of the dam, we use the Gross-Pitaevskii equation to study the quasi-one dimensional case in both perturbative ($\Delta n \ll n_{0}$) and non-perturbative ($\Delta n \sim n_{1}$) regimes. In the perturbative regime a pair of outwardly propagating dispersive wavepackets forms which can be viewed as quantum versions of undular tidal bores that have analytic forms at long times in terms of the integrals of Airy functions with a wavelength that grows as $(\hbar^2 t)^{1/3}$. Airy functions are the universal wave functions that dress structurally stable fold caustics where pairs of rays coalesce, and, indeed, we show that the quantum dam break problem has the same ray structure as the naturally occurring caustic phenomenon of a double rainbow, including Alexander's dark band between the two bows where no light is scattered: we identify the intermediate density plateau in the dam break as an analogous `silent band' where only evanescent sound waves can exist. In the opposite regime of a non-perturbative dam break we show that a self-induced sonic horizon occurs when $\Delta n \geq (8/9)n_{1}$. A discussion of possible experimental schemes for amplification of quantum undulations is included as well as an alternative scheme for dispersive wave generation where a constant flow is imprinted on a BEC in box trap.

The Hamilton-Jacobi [HJ] analysis and the Faddeev-Jackiw [FJ] quantization for the linearized, non-projectable $\lambda R$ model, extended by the lowest-order term of Blas, Pujolàs, and Sibiryakov and the higher-order term involving the square of the Cotton tensor, were carried out. The cases $\lambda \neq \frac{1}{3}$ and $\lambda = \frac{1}{3}$ are analyzed. Within the Hamilton-Jacobi framework, we obtain the complete set of Hamiltonians, generalized brackets, fundamental differential, characteristic equations, and gauge transformations of the theory. In the Faddeev-Jackiw framework, a symplectic tensor is constructed from which the fundamental brackets are identified. The HJ and FJ brackets coincide, and from the symplectic tensor, the measure for the quantum path integral is calculated. The differences between the cases $\lambda \neq \frac{1}{3}$ and $\lambda = \frac{1}{3}$ at the quantum path integral are discussed.

In this paper, we develop a mixed quantization technique for graph vector bundles that combines semiclassical analysis and quantum cat maps. We apply it to study the Kesten–McKay law and quantum ergodicity.

We give a finite-temperature test for the stability of low-energy spectral subspaces. The test uses a local determinant defect for the alpha-z Renyi kernel of two Gibbs states and bounds the tracial L2 distance between their low-energy projections. The main tool is a spectral-flattening principle for bounded positive injective operators in a semifinite von Neumann algebra. It reduces the local product defect to an exact two-projection formula. The thermal result applies to lower-bounded affiliated Hamiltonians with trace-class Gibbs operators and does not require the Hamiltonians to be close in operator norm. We give an infinite-level example with unbounded Hamiltonians. For matrices, the result gives a sum of principal-angle remainders. For qubits, the defect and the thermal gaps determine the ground-state overlap exactly. The same estimate gives a uniform error bound for bounded observables in an effective low-energy model. We also prove the determinant inequalities, their equality cases, and z-monotonicity in matrix, finite, and semifinite settings.

Confined cell migration is governed by nuclear and cellular deformation. In this work, we formulate a two-dimensional mathematical framework in which the cell and nuclear membranes are represented as evolving closed surfaces governed by force-balance laws incorporating surface tension, bending elasticity, and external mechanical interactions. Nuclear mechanics depends on several factors, including chromatin organization and compaction, which are regulated by epigenetic modifications. Molecular regulation at the nuclear periphery is described by a surface reaction-diffusion system posed on the evolving nuclear surface. We apply this coupled framework to an experimentally characterized lung cancer cell model in which the balance between the enzymes SETDB1 and SUV39H1 regulates the peripheral deposition of H3K9me3. Using experimentally measured mechanical parameters, we represent control (Ctrl) and SETDB1 knockout (SETDB1 KO) conditions, which display distinct H3K9me3 distributions, especially at the nuclear periphery. Numerical solutions obtained using the evolving surface finite element method qualitatively reproduce the changes in nuclear mechanics as well as part of the experimentally observed increase in the entry time in SETDB1 KO cells. The model therefore reproduces the direction of the experimental observations, while the residual difference between simulated and measured entry times indicates that the cellular response is multifactorial.

We study braid representations obtained from character-twisted Burau inputs by the Katz-Long-Moody (KLM) construction, which corresponds to multiplicative middle convolution for KZ-type equations. For $n\ge2$, let $q=e^{2\pi ia}$, $\tau=e^{2\pi ic}$, where $a,c>0$ and $nc+(n+1)a<1$. Known form and signature results give a positive-definite invariant form for $\lambda=e^{2\pi il}$, $0<l<1-nc-(n+1)a$. For fixed input, unitary realizations are unique up to simultaneous unitary conjugation. We address two questions. First, we determine spectra and invariant summands of the $M=n(n+1)$-dimensional output. The generator spectrum recovers $(q,\tau)$. The output decomposes into two trivial and two reduced Burau summands, and an $n(n-1)$-dimensional summand isomorphic to Long's reduced construction, generically irreducible for $n=3,4$. If $a$ or $c$ is irrational, every generator is non-Clifford in every finite-dimensional Weyl-Heisenberg frame; if $1,a,c$ are linearly independent over $\mathbb{Q}$, every generator is entangling for every nontrivial bipartition. The full braid image is not projectively dense. For general positive-definite unitary inputs, irreducibility of the full braid output at a nonexceptional parameter forces the free-group input operators to commute. Second, we construct one orthonormal basis giving each generator one block of size $2(n+1)$, $n-2$ blocks of size two, and identity elsewhere. Each generator is a product of $2n$ rank-one phase reflections and, in a fixed binary encoding, an $O(M)$-length word in one-qubit unitaries and CNOTs, without auxiliary qubits and including direction preparation and its inverse. We give finite-alphabet approximation, conditioning and error bounds, and a matched-resource comparison with a Quantum Shannon reference compiler. The gate bound is exponential in the encoded qubit count and excludes classical preprocessing.

The quantum baker map is a standard model of quantum chaos, but rigorous analysis of how it spreads random perturbations remains difficult. Following the repeated “baker plus refreshed diagonal perturbation” architecture of Schack and Caves, we study a tractable strong-noise model based on the Balazs–Voros quantization: at each time step, its unitary propagator is followed by a diagonal unitary whose entries are independent and uniform on the unit circle. The resulting random propagator is a structured random walk on $\U(N)$, $N=2^k$. We quantify how quickly the baker dynamics spreads this basis-local randomness at three levels: ensemble means, two-copy statistics, and the full law of the accumulated propagator. The walk becomes an exact unitary $1$-design at time $k+1$, while its fixed-accuracy $2$-design mixing time is $\Theta(\log N)$. Thus the induced state ensembles reproduce Haar means and two-copy statistics on the Ehrenfest scale. The full law mixes much more slowly. A phase-adapted path coupling gives normalized Wasserstein mixing in $O_\varepsilon(N)$ steps, whereas metric-entropy and Haar small-ball estimates give an $\Omega_\varepsilon(N/\log N)$ lower bound and asymptotically maximal distance at every $t=o(N/\log N)$. The law remains singular with respect to Haar measure for every $t<N$. The moment results and the Wasserstein upper bound are driven by the unistochastic Markov kernel associated with $B_N$, an affine dyadic chain whose nonconstant Fourier modes vanish exactly after $k$ steps. The resulting hierarchy separates Ehrenfest-scale randomization of state ensembles from near-Heisenberg-scale exploration of the full unitary group.

We study how thermal equilibrium states of infinite fermionic lattices respond to perturbations of the Hamiltonian. Our main result concerns extensive perturbations of a chosen KMS state. We show that exponential decay of correlations and exponential local indistinguishability under the extensive perturbation yield stretched-exponential decay of correlations for the perturbed state. Neither the initial nor the perturbed state is required to be unique, allowing stability to be studied in the presence of phase coexistence. Our approach uses Quantum Belief Propagation (QBP), which we establish directly on the underlying $C^*$-algebra. Combining QBP with Lieb–Robinson bounds, we show implications between decay of correlations, the principle that local perturbations perturb locally, and local indistinguishability, closing the cycle for perturbations of the tracial state under the stated decay, support and uniformity assumptions. For extensive perturbations under uniform clustering conditions, we also obtain a differentiable path of local equilibrium expectations and bound their change linearly in the interaction norm of the perturbation.

We derive explicit, absolutely convergent series formulas for the quantum, unassisted classical, and entanglement-assisted classical capacities of the qubit amplitude damping channel, eliminating residual optimizations and implicit roots. We also obtain analytical upper and lower bounds on two-way assisted quantum and private communication capacities by evaluating a balanced-squashing bound and optimizing the reverse coherent information. Our results characterize the optimal input populations, establish convergence and truncation properties, and provide analytical benchmarks for quantum and classical communication over dissipative channels.

The quantum Cramér-Rao bound constrains the precision of parameter estimation through the symmetric logarithmic derivative quantum Fisher information (SLD QFI). It is asymptotically achievable for regular single-parameter estimation models, but generally not in the multiparameter setting, where optimal measurements for different parameters may be incompatible. For multiparameter estimation, allowing collective measurements, the corresponding asymptotically achievable precision limit is the Holevo bound, whose standard formulation is an optimization over auxiliary Hermitian operators. We bridge this apparent difference in formulation by giving an exact characterization of the Holevo bound in terms of a family of QFIs interpolating between the SLD and right logarithmic derivative (RLD) QFIs. Specifically, for any locally identifiable finite-dimensional estimation problem, we prove that the operator-feasible region in the standard variational definition of the Holevo bound coincides with the intersection of the regions defined by the corresponding Cramér-Rao-type matrix constraints for the entire QFI family. This characterization also applies to rank-deficient states and does not require any prior choice of weight matrix. Consequently, the conventional weight-dependent Holevo bound is recovered by minimizing the corresponding weighted cost over the resulting common feasible region.

Sipser-Spielman codes and Dijkgraaf-Witten twisted gauge theories are among the most profound ideas in the last few decades in the areas of computer science and mathematical physics respectively. This work unifies them in the same framework using the language of sheaf cohomology, and produces a new framework of twisted sheaf gauge theories describing a class of quantum low-density parity-check (qLDPC) codes with non-Abelian $D_4$ topological order. The corresponding twisted Clifford-stabilizer qLDPC code can be obtained from gauging a new type of 0-form sub-complex symmetry from a qLDPC code defined on a general sheaf complex, and can be interpreted as a topological defect network of non-Abelian $D_4$ patches glued together with proper gapped interfaces. As an application, one can use this to realize a magic state fountain via the gauging measurement of the addressable logical CZ gates as 0-form subcomplex symmetries in a 2D hypergraph-product (HGP) sheaf code. This includes a scheme of subdividing an arbitrary constant-rate 2D HGP code with parameters $[[n,\Theta(n), \Omega(n^{1/2})]]$ into a quantum sheaf code which allows preparation of $\Theta(n^{1/2})$ logical CZ magic states in parallel, equivalent to the recent geometric construction using the code-to-manifold mapping in (arXiv:2601.06736). Moreover, using the recent sheaf complex and algebraic code constructions by Golowich-Tamo-Zhu (arXiv:2609.27801) with parameters $[[n,\Theta(n^{1-\epsilon}), \Omega(n^{(1-\epsilon)/2})]]$ for arbitrary small $\epsilon$, one can prepare $\Theta(n^{1-\epsilon})$ CZ magic states in parallel and hence achieve an almost-constant magic rate.

We study first-order differential quantities built from solutions of second-order black-hole wave equations. These quantities can satisfy new differential equations that contain extra singular points, even when the original physical solutions remain regular. We show that an isolated zero of the transformation determinant can produce such an apparent singularity, but it does not by itself create a new resonance or quasinormal-mode frequency. We then study the corresponding Green functions and show that the physical transformed response has the same spectral poles as the original problem, provided that the transformation does not become globally degenerate. For the Regge–Wheeler and Zerilli equations, this means that ordinary turning points can appear as singular points of the derivative equation without changing the nonzero-frequency quasinormal-mode spectrum. The zero-frequency case is different and must be studied separately.

We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a multi-well potential and $\varepsilon$ is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted $\ell^2$-estimates for the difference of these and the exact eigenfunctions of the associated Dirichlet-operators.

This Letter transitions neutrino mixing from an empirical fit to a geometric law by analytically deriving the PMNS mixing angles and mass spectrum from an $A_5$ topological manifold. Solving the Quantum Yang-Baxter Equation forces a geometric fracture at $M_0 \approx 3.88 \times 10^{16}$ GeV. Projecting this to the Electroweak scale yields parameter-free targets: maximal CP violation $\delta_{CP} = \pm \pi/2$, a mass sum $\sum m_i \approx 72.48$ meV, with $m_1 < m_3$, and effective masses $m_\beta \approx 12.47$ meV and $m_{\beta\beta} \approx 8.20$ meV, setting definitive experimental thresholds.

In this work, we define $(3+1)$-dimensional topological quantum field theories (TQFTs) and their lattice realizations with input data given by a pivotal fusion 2-category; more precisely, our framework applies in wider generality to presemisimple locally fusion pivotal tensor $2$-categories, which are equipped with a so-called shadow trace. We introduce a decoration of triangulated manifolds by the input data, and develop a $20j$-symbol formalism for computing 4-simplex scattering amplitudes. We verify the topological invariance of 20$j$-symbol summations under the 4-dimensional Pachner moves. Furthermore, we also construct explicitly a 3-dimensional Levin-Wen-type lattice Hamiltonian model realizing the state-sum invariant. Our construction can be understood as a higher-dimensional analogue of the spin-networks construction for the Turaev-Viro invariant. Examples arising from $2$-groups are discussed from the perspective of higher gauge theory.

We prove that on the Sierpinski gasket, a prototype of compact fractal space, Dirac distributions arise as the weak limit of probability measures associated with sequences of high energy eigenfunctions of the Laplacian in a way that cannot happen in compact Riemannian manifolds.

We study two hierarchies of bivariate generating functions defined by differential recursions: a self-dual hierarchy and a reduced-symmetry hierarchy. Their rational coefficient functions produce palindromic and anti-palindromic triangles, including OEIS A336110 and A140136. For general differential order $p$, the reduced hierarchy decomposes into $p-1$ Horn-type companions. An explicit differential intertwiner of order $p-3$ relates the last companion to the normalized self-dual generating function. The companions share a principal symbol and a rationally parametrized characteristic curve with a projective $S_3$ action. At $p=2$ this construction yields the Narayana quadratic. At $p=3$ the symmetry lifts to $S_4$: two algebraic sectors are edge resolvents of one quartic root configuration, and the nonsymmetric Gross–Witten–Wadia generating function has degree four. For $p=4$ and $p=5$ we derive algebraic equations of degrees six and eight, respectively, and find Galois groups $S_6$ and $S_8$ at generic specializations. At $p=6$ we verify a degree-ten specialization with Galois group $S_{10}$. Their discriminants exhibit a universal cubic characteristic factor and a squared lower-order factor. A boundary factorization explains the observed algebraic degrees and discriminant monomials. We formulate conjectures on algebraicity and Galois groups in the reduced hierarchy and identify a possible mechanism for transcendence in the higher self-dual hierarchy.

Stochastic processes underpin dynamics across biology, physics, epidemiology, and finance, yet accurately simulating them remains a major challenge. Classical approaches such as the Gillespie algorithm are exact for Markovian, time-independent systems, where propensities depend only on the current state and agents of a given type are statistically identical. While efficient, this framework misses a defining feature of many real systems: heterogeneity and memory at the level of individual agents. Cells may divide or differentiate on distinct intrinsic timescales, individuals may preferentially interact with specific partners, and inter-event-time distributions can deviate strongly from the exponential. We introduce MOSAIC (Modeling of Stochastic Agents with Individual Complexity), a general and scalable framework that embeds agent-specific properties directly into the dynamics. MOSAIC unifies heterogeneous rates, dynamic interaction preferences, and both Markovian and non-Markovian waiting-time distributions within a single stochastic formalism, while retaining Gillespie-like computational cost.

Quantum algebras ($q$-algebras) have been used in the literature to deform spin-chain models. In particular, in previous works by the present authors, the Kittel–Shore Hamiltonian was deformed using $\mathfrak{su}_q(2)$ algebra. The deformed Hamiltonian was presented for a general spin, and the thermodynamic properties were studied for spin-1/2 particles. In this paper, we focus on the $q$-deformation of the Kittel–Shore model for spin-1 systems. We analyse the impact of this deformation on the thermodynamic properties and phase transitions of the system as a function of the deformation parameter $q$. Specifically, we study the specific heat, magnetic susceptibility, and magnetisation, including a detailed analysis of the Curie temperature characterising the ferromagnetic phase. Furthermore, a finite-size scaling analysis is conducted for both the ferromagnetic and antiferromagnetic cases. Results are systematically compared with those of the undeformed model to reveal the physical effects of the $q$-deformation.

This study derives the asymptotic conditions under which the massless stretched string formulation is valid for the equivalent dynamics of the inertial string carrying a moving point mass. Consequently, the governing equation under such conditions can be written as a singularly perturbed partial differential equation when the inertia of the string is included. When the inertia of the string is ignored, the governing equation reduces to a non-homogeneous hypergeometric ordinary differential equation governing the transverse motion of the moving mass. This hypergeometric equation has a discontinuous solution at the terminating boundary known as the trajectory paradox. However, numerically solving the actual equation yields a continuous solution with a rapid variation near the boundary. The complete solution resembles the inner solution as well, whereas the hypergeometric ordinary differential equation leads to an outer solution, revealing that the trajectory paradox is a boundary layer.