We study the action of the Hecke triangle groups $G_q$ on $λ_q \mathbb{Q}(λ_q^2) \cup \{\infty\}$ with $λ_q = 2 \cos (π/ q)$. When $q = 18$, we show the existence of infinitely many distinct orbits of fixed points of special hyperbolic elements of $G_q$. We also find new orbits for several other values of $q$. These results provide new examples of special affine pseudo-Anosov homeomorphisms on the unfoldings of regular $q$-gons. In particular, on the unfolding of the regular $18$-gon, there are infinitely many distinct Veech group orbits of directions invariant under a special affine pseudo-Anosov.

We show that any big line bundle on a smooth projective variety admits a special Fujita approximation: the volume and the first Riemann-Roch coefficient are both approximated by those of ample $\mathbb{Q}$-line bundles on higher models. Exploiting previous works by Boucksom, Jonsson and Li, we solve the Boucksom-Jonsson Regularization Conjecture on the Non-Archimedean entropy functional. As a main consequence, we obtain a solution to the (uniform version of the) Yau-Tian-Donaldson Conjecture: a polarized smooth projective variety $(X,L)$ admits a cscK metric if and only if it is $\mathrm{Aut}^\circ(X,L)$-uniformly $K$-stable. This extends the known Yau-Tian-Donaldson correspondence for smooth Fano varieties.

We investigate the metal-insulator transition driven by the onsite repulsive interaction $U$ in an altermagnetic Hubbard model defined on a Lieb lattice. Using the slave-rotor approach at half filling, we find that the system exhibits a cascade of interaction-driven phase transitions. As $U$ increases, the system evolves from a normal metal to an altermagnetic metal, then to an altermagnetic insulator, and eventually to an altermagnetic Mott insulator characterized by the complete suppression of the quasiparticle weight. These phases are supported by the calculation of the electronic spectral function, which features spin-split bands in both the metallic and insulating regimes. However, the spin splitting becomes substantially suppressed in the Mott insulating phase. Our results suggest that the observation of spin splitting in the spectral function of $d$-wave altermagnets with a Lieb-lattice-like structure may be limited to the weak-to-moderate correlation regime.

Weakly ionized plasmas can be found in the lower layers of the solar and stellar atmospheres and in structures such as prominences and spicules. A variety of density perturbations and bulk flows detected in these environments have been explained as the result of the ponderomotive force generated by nonlinear Alfvénic waves. In addition, the dissipation of the energy carried by these waves leads to heating of the plasma. Here, we use a two-fluid model to study the combined influence of Hall's current and elastic collisions between ions and neutrals on the propagation of linearly and circularly polarized transverse waves in weakly ionized plasmas. We derive analytical expressions for the damping and heating rates, showing their dependence on the strength of the collisional coupling and on the polarization state. We also perform numerical simulations to investigate the nonlinear generation of density perturbations and bulk flows related to the ponderomotive force and the energy dissipation by the ion-neutral interaction. We find that the nonlinear perturbations associated with the circularly polarized eigenmodes do not show the oscillatory motions typically caused by linearly polarized eigenmodes, but they retain the non-oscillatory bulk flows. We also briefly discuss how in weak coupling conditions the nonlinear dynamics of the neutral fluid is mainly driven by the wave energy dissipation while the ponderomotive force only directly acts on the charged fluid, resulting in different amplitudes of the longitudinal motions and the perturbations of density and temperature.

Importance sampling (IS) consists in biasing samples from a distribution $f$ towards another distribution $g$. Concretely, given samples $X_i$ from $f$, the IS measure is $$\hat{g}_n = \frac{1}{Z_n}\sum_{i=1}^n \frac{g(X_i)}{f(X_i)} δ_{X_i},$$ with $Z_n = \sum_{i=1}^n \frac{g(X_i)}{f(X_i)}$. The random measure $\hat{g}_n$ approximates $g$, and is used in many contexts ranging from Monte Carlo integration to Bayesian inference. We show that, in high dimension ($d \geqslant 3$), the Wasserstein cost $W_p^p(\hat{g}_n, g)$ has order $n^{-p/d}$ in expectation, i.e. $$β^{\mathrm{low}}_{p,d}\int gf^{-p/d}\leqslant \liminf_{n \to \infty} n^{p/d} \mathbb{E}[W_p^p(\hat{g}_n, g)] \leqslant \limsup_{n \to \infty} n^{p/d} \mathbb{E}[W_p^p(\hat{g}_n, g)] \leqslantβ_{p,d} \int g f^{-p/d}$$ where $0<β^{\mathrm{low}}_{p,d}\leqslant β_{p,d}$ are constants depending only on $p$ and $d$, which are equal for $p=2$ and conjectured to be equal for any $p\geqslant 1$. Our results are valid for all $p\geqslant 1$ and $d\geqslant 3$. In the case where $β^{\mathrm{low}}_{p,d} = β_{p,d}$, we show that the asymptotically optimal sampling distribution $f^*$ for importance sampling is not equal to $g$ but to a tempered version of $g$, namely $f^* \propto g^{d/(p+d)}$, which is reminiscent of Zador's theorem in the domain of measure quantization.

Connected Submodular Maximization (CSM) is a graph problem with important applications to wireless network deployment, path planning, epidemic outbreaks, and cancer genome studies. In CSM, we are given a graph $G$, a non-negative monotone submodular function $f$ on subsets of the vertex set of $G$, and an integer $k$. The goal is to select a tree in $G$, with $k$ edges, whose vertex set maximizes $f$. We also study the more general Directed and Directed Rooted variants of CSM (DCSM and DRCSM respectively). In both variants, $G$ is directed and the solution must be an out-tree in $G$, with $k$ edges, whose vertex set maximizes $f$; DRCSM further specifies a vertex to be the root of the selected out-tree. For CSM, several previous works have proposed polynomial time approximation algorithms; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{\sqrt{k}})$-approximation. We can also parameterize the approximation factor by the radius of the optimal solution, denoted by $r$; the state-of-the-art polynomial time algorithm achieves a $Ω(\frac{1}{r})$-approximation. In this paper, we improve on the state-of-the-art approximation factor for CSM with respect to $r$ as well as $k$, noting that $r \leq k$. We propose a polynomial time framework that, for (Directed) CSM, achieves a $Ω(\frac{\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation for every constant $\varepsilon \in (0, 1]$. For DRCSM, our framework achieves a $Ω(\frac{δ\varepsilon^{3}}{{r}^{\varepsilon}})$-approximation that violates the size constraint by at most a factor of $1 + δ$ for every $δ\in [\frac{1}{k}, 1]$. A key component of our framework is GreedyRadius, which is an algorithm for DRCSM that takes another algorithm with a bicriteria approximation factor in terms of $k$ and outputs a solution with the same bicriteria approximation factor (up to constants) in terms of $r$.

In material systems with slab geometry, the surface contribution to physical responses is commonly expected to diminish rapidly with increasing thickness, giving way to the bulk response. Here, we show that this conventional wisdom is violated in a class of gate-induced responses, including gate-induced orbital and spin magnetization as well as cross-field anomalous thermoelectric transport. We develop a general framework for these effects, which naturally decomposes the total response into surface- and bulk-contributions treated on equal footing. Remarkably, the volume-averaged surface contribution remains finite in the thick-slab limit and exhibits the same thickness scaling as the bulk term. Furthermore, the surface response originates from band geometric quantities distinct from those in the bulk, being constrained solely by surface symmetries. As a result, it can dominate the overall response when the bulk contribution is symmetry-forbidden. Taking MnBi$_2$Te$_4$ multilayers as an example, we predict a strong surface-dominated cross-field anomalous Nernst effect arising from surface Berry curvature, which is readily accessible to experimental detection. These findings reveal a previously overlooked significance of surface response and open a new direction in the study of surface quantum geometry.

We review known linear and matrix generalizations of Hall's classic ``marriage theorem'' and Kőnig's theorem on partial matchings in bipartite graphs, and relate them to linear and matrix generalizations of Dilworth's theorem about chains and antichains in posets and Menger's theorem about disjoint paths in directed graphs.

We identify chiral inversion points in slow-light, glide-plane-symmetric, photonic-crystal waveguides, defined as fixed locations where the local optical chirality changes sign over a narrow wavelength range. We experimentally access this behaviour using a waveguide-embedded InAs/InGaAs quantum dot. The slow-light spectral region is determined from time-integrated and time-resolved photoluminescence, and the dot exciton is electrically tuned across the slow-light bandwidth via the quantum-confined Stark effect. As the emission wavelength is swept through the slow-light region, the directional emission contrast shows a strong wavelength dependence and a sign reversal, consistent with the identified chiral inversion point. Numerical simulations attribute the switching primarily to the pronounced spectral variation of the local optical chirality for emitters displaced from the waveguide center. These results demonstrate on-demand electrical switching of chiral light-matter coupling in nanophotonic waveguides and enable tunable chiral interfaces for integrated quantum photonic devices.

The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such singularities violate the long-wave assumptions underlying the model, their appearance limits the regime of validity of the equations. In this work, we introduce a regularized shallow-water system in which the nonlinear terms are modified by a bounded operator. This regularization removes the standard derivative-steepening mechanism responsible for shock formation in the classical system while remaining consistent with the long-wave regime. We establish local well-posedness and small-data global well-posedness in Sobolev spaces that exclude singularity formation. In addition, numerical simulations indicate that the system admits solitary-wave solutions.

The outer rings (ORs) of Supernova (SN) 1987A were ejected ~20000 years before the explosion. Their characterisation is crucial for constraining the properties of the progenitor of this famous SN. While numerous studies investigated in detail the ejecta, equatorial ring (ER), and reverse shocks, few were dedicated to the ORs. We fill this gap and investigate the ORs physical properties. We analyse data obtained over a long temporal period, from multiple instruments, and over a wide wavelength range from optical to mid-infrared of the northern and southern ORs (NOR and SOR). We combine observations taken with HST between 1994 and 2022, VLT/MUSE in 2023, and JWST in 2022 and 2024. We measure emission flux in the ORs in HST and JWST/NIRCam images. We extract optical and mid-infrared spectra for the ORs in MUSE and JWST/MIRI/MRS data and measure line emission fluxes. We analyse the evolution of the ORs clumps' morphology over time with HST. The optical lightcurves of the ORs have shown a steady decline with time over the last 30 years. It is expected as the ORs were ionised by the initial SN UV-flash and are since then fading. The observations do not show any sign of interaction of the SN ejecta with the ORs. We estimated the decay times for [O III] to be 900 and 680 days for the NOR and SOR, and for Halpha+[N II] to be 15870 and 7160 days for the NOR and SOR. We constrained the temperature from the optical [N II] lines to 13400-16900K and 11800-14500K for the NOR and SOR. We constrained the electron density from the optical [S II] lines to 610-670cm-3 and 720-790cm-3 for the NOR and SOR. The spectra of the ORs differ significantly from the spectrum of the ER in lines detected and line ratios. The ORs will likely keep on fading for the next years, until the SN ejecta sweep them up. Continued monitoring of SN1987A and its ring system at all wavelengths is essential to capture this instant.

Externally seeded free-electron lasers (FELs) are promising approaches for generating fully coherent soft-X-ray radiation. Their extension to shorter wavelengths and MHz-level repetition rates is, however, constrained by the limited availability of high-repetition-rate seed lasers with sufficient energy modulation. Recent self-amplification and direct-amplification experiments at the Shanghai Soft X-ray FEL facility have significantly relaxed the peak-power requirement for high-gain harmonic generation (HGHG) and opened a practical path toward echo-enabled harmonic generation (EEHG). Using the SHINE bypass line, three compatible high-repetition-rate seeded-FEL configurations are explored: self-modulation cascaded HGHG, self-modulation EEHG, and direct-amplification-driven EEHG. Numerical simulations indicate that these schemes can provide flexible routes toward MHz-level operation with harmonic generation beyond the 30th order. A common modulator-chicane layout is proposed to preserve compatibility among the candidate modes and to support future optimization and experimental implementation at SHINE.

A novel mixed spectral-Galerkin method based on generalized ball polynomials is proposed for solving the biharmonic equation on a unit ball. By introducing an auxiliary variable to decouple the biharmonic equation into a system of second-order equations, the corresponding discrete scheme yields a strictly diagonal stiffness matrix, which significantly enhances the computational efficiency. Rigorous a-priori error estimates are established to demonstrate the exponential convergence rates in both the $L^2$- and $H^1$-norms. Extensive numerical experiments are conducted to verify the theoretical analysis and confirm the high efficiency and accuracy of the proposed scheme.

Current observations with the James Webb Space Telescope (JWST) suggest that star-forming galaxies produce enough ionizing (LyC; $λ< 91.2$ nm) photons to drive cosmic reionization, but the efficiency with which these photons escape their host galaxies remains uncertain. Absorption by the neutral intergalactic medium progressively suppresses direct LyC detections above redshift $z\sim3$, forcing astronomers to rely on indirect diagnostics of LyC escape calibrated at low redshift. Low-resolution ultraviolet observations of high-redshift analogs obtained with the Cosmic Origins Spectrograph onboard the Hubble Space Telescope (HST) have been critical for developing these diagnostics. These studies suggest that stellar feedback plays a central role in regulating LyC escape, although the role of galactic winds and the underlying physical mechanisms remain poorly constrained. High-resolution spectroscopy blueward of 160.0 nm (rest-frame) is required to resolve the kinematic structure of the winds and reveal the physics governing LyC escape. Such observations are currently only possible with HST and represent a major science driver for the future Habitable Worlds Observatory (HWO). Extending the lifetime of HST and prioritizing ultraviolet observations are essential for interpreting current JWST studies of the early Universe and important preparatory science for HWO.

Contact (or mixing, or more generally connectivity) matrices are a fundamental component of modelling and inference for infectious disease epidemiology. Their structure and parametrisation directly accounts for the frequency of interactions between different subpopulations of individuals, as well as having the potential to encode dynamic heterogeneity in these interactions across demographic axes, space and time. Considerable research has been devoted to the structure and estimation of (components of) these matrices to help inform outbreak control and forecast disease spread. In this paper, we review the existing literature on the data types used to construct contact matrices and the methods for incorporating uncertainties and heterogeneities into them. We also highlight remaining challenges and future directions in the use of these contact matrices for epidemiological research.

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $Ω(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erdős, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

Driven-dissipative qubit-resonator dynamics, which are the basis of most dispersive superconducting qubit measurement schemes, are often modeled with Lindblad master equations built from subsystem local jump operators, even when the qubit and resonator are appreciably hybridized. In this work we revisit this setting using a microscopic Bloch-Redfield approach, where dissipation is constructed in the eigenbasis of the coupled qubit-resonator Hamiltonian with a complete, frequency dependent, open system description of the transmission line environment. Here, we show that the Lindblad and Bloch-Redfield decay rates can be quantitatively different in the absence of driving, while in the driven case we demonstrate that the time-independent Redfield dissipator and its time-dependent generalization can show qualitatively different behaviors as a function of driving strength. Finally, we investigate the effects of driving in structured spectral densities, recovering the suppression of measurement-induced relaxation in the presence of a so called Purcell filter.

Using the Baxter-Kelland-Wu coupling and the convergence of the height function of the six-vertex model to the Gaussian Free Field, we extract critical exponents for the planar critical random-cluster model at $q=4$, and the planar four-state Potts model.

Noise is usually regarded as the main obstacle to achieving a scalable quantum advantage, but recent evidence in quantum reservoir computing [L. Domingo, F. Borondo, and G. G. Carlo. Taking advantage of noise in quantum reservoir computing, Scientific Reports, 13:8790, 2023] suggests that certain channels can, in appropriate regimes, improve performance by enriching the reservoir's effective dynamics. Motivated by this idea we propose a geometric mechanism to explain how non-unital noise applied together with a universal gate set leads to a faster approach to Haar-like distributions of the final states. We find that noise of this kind induces an effective volume expansion on the manifold of pure states. In order to intuitively understand this we use a minimal 1 qubit model where we take the amplitude damping channel and combine it with a renormalization rule that associates to each resulting mixed state a representative pure state. This composition defines a globally expanding nonlinear map on the space of pure states. We analytically derive the local area expansion factor and identify the global expansion threshold. Finally, we combine amplitude damping with the G3 = {H, T, CNOT} universal gate set to show how the approach to Haar-like behavior is faster in an appropriate parameter region. This leads us to propose noise as a possible resource in future quantum algorithms.

The most promising candidate for short-duration gamma-ray bursts (GRBs) is the merger of two neutron stars (NSs), which produces kilonovae (KNe) in the aftermath. This merging can result in a fast-spinning, highly magnetic NS, known as a millisecond magnetar, whose accretion processes can explain different phases in GRBs. The identification of a KN associated with the atypical long-duration GRB 211211A contradicted the classification schemes of the GRB progenitors. This study presents a comprehensive analysis of gamma- and X-ray observations, focusing on modeling X-ray data from a long-lived magnetar with two distinct fallback accretion rates ($\dot{M}\propto t^0$ and $\propto t^{\frac12}$) during the initial phase. The internal energy dissipation of the magnetar spin-down power, through the magnetization parameter, accounts for the long duration of the prompt gamma-ray episode observed in GRB 211211A. Furthermore, we provide a satisfactory explanation for the precursor and extended emissions following the prompt episode within the magnetar model with two fallback accretion rates. Although these accretion rates explain different characteristics, the model that incorporates a variable accretion rate offers a more accurate description. The current scenario for the GRB 211211A observations aligns with a compact binary merger that produces a long-lived magnetar instead of an immediate black hole.

Standard EU energy system modelling approaches optimize for least-cost, leading to highly centralized systems, in conflict with political feasibility and physical security concerns. This paper incorporates decentralisation as a constraint in a European energy system model using a novel, linear load-weighted renewable capacity constraint, the K-parameter, which scales with total system renewable capacity to avoid interference with decarbonisation targets. The model is a 37-node electricity-only brownfield system based on the PyPSA-EUR framework, with projected 2050 loads and technology costs. A total of 105 optimized scenarios are analyzed at 14 levels of decarbonization and 8 levels of decentralization. Full decarbonization leads to an 80% cost increase due to, among other factors, a 78% increase in energy generation capacity. Without decentralisation constraints, system equity initially improves but collapses at high decarbonisation levels due to concentration in regions with optimal renewable resources. Moderate decentralization of K=7 achieves 76% of the equity benefits at only a 9% cost increase compared to K=1. This indicates that moderate decentralization can be a viable strategy to balance societal preferences and cost-efficiency in the European energy transition.

The Atacama Large (sub-)millimeter Array (ALMA) has been in scientific operations for almost 15 years. We celebrate this achievement by providing a summary of the ``Disks and planet formation'' scientific category, with an emphasis on the disks located in the nearby star-forming regions. As of the beginning of February 2026, ALMA had observed 3933 independent coordinates, which we analyzed by their location in the sky, frequency coverage, exposure time, spectral line coverage, and angular resolution. We encourage the community to explore new scientific questions that are made possible through the archival datasets.

Molecular emission is observed in a wide variety of astrophysical environments, yet a substantial fraction of spectral features detected at mm wavelengths remains unidentified. Identifying these features is essential for constraining the inventory of interstellar and circumstellar molecules and for understanding the chemical pathways operating in evolved stars. In this study, we investigate a set of weak unidentified molecular features detected in the C-rich PN IC 418. We aim to constrain the nature of their molecular carriers and assess their possible connection to fullerene-related chemistry. High-sensitivity observations at 2, 3, and 7 mm were carried out using the RT40m and IRAM 30m radio telescopes. The spectral features found in these data sets were compared with public molecular spectroscopic databases. Spectral-pattern searches and line fits were performed under linear, symmetric, and asymmetric rotor approximations to estimate rotational constants of potential molecular carriers. We report the detection of 20 weak (SNR~2-13) unidentified features (UFs), none of which can be attributed to mm radio recombination lines, instrumental artifacts, or known molecular species. The observed features are inconsistent with the regular line spacing expected from linear molecules or symmetric rotors, although a recurrent doublet-like pattern is identified. The estimated rotational constants (B~2 500-3 660 MHz) suggest carriers with 4-13 atoms and are incompatible with those expected from C60 derivatives (B~100 MHz). Non-planar carbonaceous molecules produced during the destruction/processing of hydrogenated amorphous carbon (HAC) grains are suggested as promising candidates. The catalog of UFs presented here is made publicly available to facilitate future comparisons with laboratory measurements and theoretical calculations aimed at identifying their molecular carriers.

We use the machinery of a conditional probability space (Rényi, 1955) to obtain an Agreement Theorem (Aumann, 1976) under general conditions. A conditional probability space (CPS) is a family of probability measures defined relative to a family of conditioning events that satisfies concentration and a chain rule. Using this apparatus, we derive an Agreement Theorem that dispenses with the traditional assumptions of a common prior, information partitions, positivity of measure, and knowledge operators. Our treatment can be viewed as ``deconstructing" the classic Agreement Theorem, by showing how it can be built up from local probabilistic-epistemic ingredients. The main technical contribution is to define an augmentation procedure for CPS's that adds into the conditioning family all (sub)events that receive probability $1$ -- thereby achieving consistency between an agent's information and subjective certainty of events.

Generating and characterizing uniform and staggered spin polarization in antiferromagnets is one of the key challenges for antiferromagnetic spintronic technology. Here, we perform perturbative theory, group-theoretical symmetry analysis, low energy and ab initio simulations to propose that the magnetic precession near the equilibrium magnetic axis could generate finite uniform and staggered spin polarization at the opposite magnetic sublattices (referring to total magnetic and Néel vector generation) in a single AFM semiconductors. This response does not require the heterojunction setup and could eliminate the lattice mismatch issues at the junction. Through scrutinizing all symmetrically-protected vanishing magnetic moment groups and especially focusing on parity-time (PT ) invariant groups, we identify the symmetry constraints that describe the staggered spin accumulation responses, and disclose their fieldlike and dampinglike characters. This unravels a hidden spin accumulation mode in AFM semiconductors. Furthermore, we simulate such an effect using a perturbative approach and suggest that electric gate field and Floquet light-dressing can effectively manipulate these responses.

Electric flow sampling (elfs) is a new tool in the quantum walk toolbox and a useful primitive for solving search, sampling and optimization problems on graphs. We refine this tool by showing that there exists a zero-error transducer for implementing elfs. More broadly, we establish a zero-error transducer for reflecting about the intersection of two subspaces, yielding an errorfree transducer version of the effective gap lemma. Building on this result, we obtain improved quantum walk algorithms for estimating effective resistances and span program witness sizes with an optimal error scaling, and for sampling from the random walk arrival distribution, via the composition of many elfs. Using this last algorithm, we obtain an up-to-quadratic quantum speedup for semi-supervised learning on expander graphs.

Alloy-based perovskite solar cells offer tunable properties and improved stability, but their complexity has impeded accurate modeling, hindering development. We present a machine-learning (ML) accelerated atomistic modeling approach for the phase stability of (Cs/FA)Pb(Br/I)3 and (Cs/FA)Sn(Br/I)3 perovskites, with FA being formamidinium. To make such quaternary alloys tractable, we adopt a two-level ML strategy, combining 1) graph neural network interatomic potentials trained on density functional theory data for efficient structure relaxations with 2) secondary ML models for direct energy prediction from unrelaxed structures. These models enable computations of free energy landscapes across compositions and phases, capturing alloy disorder and FA molecular orientations. Our results reveal narrower stable composition regions for the Sn-based system compared to its Pb-based counterpart, limiting options for compositional engineering. Maximum stability occurs at high I content, and no stabilization is observed near the center of the composition space. Our results guide the design of stable perovskites.

We analyze how the interaction between local and nonlocal dispersions, combined with different types of nonlinearities, influences the smoothing effects of solutions. To achieve this goal, we consider a model that generalizes the KdV and Benjamin equations and demonstrate that its solutions exhibit Kato's smoothing effect and satisfy the propagation of regularity principle. As a result, we confirm that the higher-order dispersive term determines the local gain of fractional regularity of solutions. Our results are general; they not only recover known results for the KdV and Benjamin equations, but also provide new insights for a broader family of models of physical and mathematical interest with polynomial dispersions of arbitrary order.

All reduced descendent Gromov-Witten invariants of $K3$ and abelian surfaces in primitive curve classes can be calculated by the methods of \cite{BOPY,MPT}. To handle the imprimitive curve classes, a multiple cover formula was conjectured in \cite{ObPand} for $K3$ surfaces and in \cite{O_NLGW} for abelian surfaces. We prove here that both descendent multiple cover formulas are implied by the conjectural families GW/PT correspondence for semipositive relative 3-folds with primary insertions. The implication is proven by showing that the multiple cover formula for $S$ can be recast as a property of an appropriate localization vertex for the relative 3-fold Gromov-Witten theory of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$. The families GW/PT correspondence then transfers the multiple cover formula from the Gromov-Witten side to the stable pairs side where the formula is proven geometrically by studying cosections and applying universality properties. Along the way, we prove a DT/PT correspondence for the reduced theories of $(S\times \mathbb{P}^1/S_0 \cup S_\infty)$ using the wallcrossing techniques of Kuhn-Liu-Thimm \cite{KLT2,KLT}.

We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical background field for the quantum system. The thermodynamics of this system is contrasted with the standard Ising model. At the gapless points in the energy spectrum, we show that this system can be derived from the quantum inverse scattering method adapted to a multi-site generalization of the Yang-Baxter equation as introduced by E. Rowell and Z. Wang. The $R$-matrix is constructed using generators of extraspecial 2-groups. This helps us extract all the conserved charges and lay the framework for a general mechanism to generate such multi-site interaction spin systems that are transverse field Ising models under the hood. A remark on how to obtain P. Fendley's free-fermion in disguise models in this formalism is also included.