In this paper, we explore accelerating Hamiltonian ground state energy calculation on NISQ devices. We suggest using search-based methods together with machine learning to accelerate quantum algorithms, exemplified in the Quantum Eigensolver use case. We trained two small models on classically mined data from systems with up to 16 qubits, using XGBoost's Python regressor. We evaluated our preliminary approach on 20-, 24- and 28-qubit systems by optimising the Eigensolver's hyperparameters. These models predict hyperparameter values, leading to a 0.12% reduction in error when tested on 28-qubit systems. However, due to inconclusive results with 20- and 24-qubit systems, we suggest further examination of the training data based on Hamiltonian characteristics. In future work, we plan to train machine learning models to optimise other aspects or subroutines of quantum algorithm execution beyond its hyperparameters.

The layered cobaltate CaCoO$_2$ exhibits a unique herringbone-like structure. Serving as a potential prototype for a new class of complex lattice patterns, we study the properties of CaCoO$_2$ using X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS). Our results reveal a significant inter-plane hybridization between the Ca $4s-$ and Co $3d-$orbitals, leading to an inversion of the textbook orbital occupation of a square planar geometry. Further, our RIXS data reveal a strong low energy mode, with anomalous intensity modulations as a function of momentum transfer close to a quasi-static response suggestive of electronic and/or orbital ordering. These findings indicate that the newly discovered herringbone structure exhibited in CaCoO$_2$ may serve as a promising laboratory for the design of materials having strong electronic, orbital and lattice correlations.

We present a simple performance bound for the greedy scheme in string optimization problems that obtains strong results. Our approach vastly generalizes the group of previously established greedy curvature bounds by Conforti and Cornuéjols (1984). We consider three constants, $α_G$, $α_G'$, and $α_G''$ introduced by Conforti and Cornuéjols (1984), that are used in performance bounds of greedy schemes in submodular set optimization. We first generalize both of the $α_G$ and $α_G''$ bounds to string optimization problems in a manner that includes maximizing submodular set functions over matroids as a special case. We then derive a much simpler and computable bound that allows for applications to a far more general class of functions with string domains. We prove that our bound is superior to both the $α_G$ and $α_G''$ bounds and provide a counterexample to show that the $α_G'$ bound is incorrect under the assumptions in Conforti and Cornuéjols (1984). We conclude with two applications. The first is an application of our result to sensor coverage problems. We demonstrate our performance bound in cases where the objective function is set submodular and string submodular. The second is an application to a social welfare maximization problem with black-box utility functions.

The momentum dependence of Rashba spin-orbit coupling (RSOC) is a key ingredient for engineering topological superconductors (TSCs), yet research has overwhelmingly focused on its linear-in-momentum form. This focus has restricted time-reversal invariant TSCs to helical $p$-wave states, which are characterized by a $\mathbb{Z}_2$ topological invariant that permits at most a single Majorana Kramers pair at a given boundary. Their existence has also been tied to the stringent criterion of an odd number of Fermi surfaces (FSs). In this work, we establish higher-order RSOC as a powerful design principle to go beyond the $\mathbb{Z}_2$ classification and the odd-FS criterion. We demonstrate that a bilayer system with a pure cubic RSOC and an intrinsic odd-parity pairing on a single FS yields a rare 2D helical $f$-wave TSC. This state is characterized by a large mirror Chern number (MCN) of ${\cal N}_{\text{M}}=3$ and hosts three Kramers pairs of Majorana edge modes. Remarkably, the interplay of linear and cubic RSOCs in this bilayer can generate a helical hybrid $p+f$-wave TSC with an even larger MCN of ${\cal N}_{\text{M}}=4$ from a normal state with two FSs, thereby circumventing the conventional odd-FS criterion. Our work establishes higher-order RSOC as a "topology multiplier" for realizing TSCs with multiple Majorana Kramers channels, fundamentally reshapes the criteria for helical TSCs, and holds immediate relevance for tunable platforms like oxide heterostructures.

We investigate two-dimensional crystallization phenomena, i.e. minimality of a lattice's patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\|x-y\|)$ where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^2$ and $V:\mathbb{R}_+^*\to\mathbb{R}$ is a function. For the Heitmann-Radin sticky disk potential $V=V_{\text{HR}}$, we prove, using Brass' key result from [\textit{Computational Geometry}, 6:195--214, 1996], that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to $\|\cdot\|$. The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the $p$-norms $\|\cdot\|_p$, $p\geq 1$, which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [\textit{Arch. Ration. Mech. Anal.}, 240:987--1053] where points are constrained to be on $\mathbb{Z}^2$. Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard-Jones potential $V=V_{\text{LJ}}:r\mapsto r^{-12}-2r^{-6}$ as well as the Epstein zeta function associated to a $p$-norm $\|\cdot\|_p$, i.e. when $V=V_s:r\mapsto r^{-s}$, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to $p$.

Let $π\cln X\to Δ^m$ be a proper smooth Kähler morphism from a complex manifold $X$ to the unit polydisc $Δ^m$. Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective manifold $S$. If the canonical line bundle of $S$ is semiample, then we show that all fibers over $Δ^m$ are biholomorphic to $S$. As an application, we obtain that for smooth families where the canonical line bundle of the generic fiber is semiample, birational isotriviality is equivalent to isotriviality. Moreover, we establish a new Parshin-Arakelov type isotriviality criterion.

For every $d \geq 4$, we construct a $d$-dimensional, log canonical, $K$-trivial variety with the property that two general fibers of its Albanese morphism are not birational. This provides a strong counterexample to the Beauville--Bogomolov decomposition in the log canonical setting. This construction can also be adapted to construct a smooth quasi-projective variety of logarithmic Kodaira dimension 0 whose quasi-Albanese morphism has maximal variation. On the positive side, we show that the Albanese morphism for log canonical pairs with nef anti-canonical class is a locally stable family of pairs.

Fast Radio Bursts (FRBs), like pulsars, display radio emission from compact regions such that they can be treated as point sources. As this radiation propagates through space, they encounter sources of lensing such as a gravitational field of massive objects or inhomogeneous changes in the electron density of cold plasma. We have developed a simulation tool to generate these lensing morphologies through coherent propagation transfer functions generated by phase coherent geometric optics on a spatial grid. In the limit an FRB can be treated as a point source, the ray paths from the FRB to the observer are phase coherent. Each image will have a time delay and magnification that will alter the emitted frequency-temporal morphology of the FRB to that which is observed. The interference of these images could also decohere the observed phase properties of the images, affecting any phase related searches such as searching for the auto-correlation of the observed FRB voltage with other images in time. We present analytic test cases to demonstrate that the simulation can model qualitative properties. We provide example multi-plane lensing systems to show the capabilities of the simulation in modeling the lensed morphology of an FRB and observed phase coherence.

Gradient descent is one of the most basic algorithms for solving continuous optimization problems. In [Jordan, PRL, 95(5):050501, 2005], Jordan proposed the first quantum algorithm for estimating gradients of functions close to linear, with exponential speedup in the black-box model. This quantum algorithm was greatly enhanced and developed by [Gilyén, Arunachalam, and Wiebe, SODA, pp. 1425-1444, 2019], providing a quantum algorithm with optimal query complexity $\widetildeΘ(\sqrt{d}/\varepsilon)$ for a class of smooth functions of $d$ variables, where $\varepsilon$ is the accuracy. This is quadratically faster than classical algorithms for the same problem. In this work, we continue this research by proposing a new quantum algorithm for another class of functions, namely, analytic functions $f(\boldsymbol{x})$ which are well-defined over the complex field. Given phase oracles to query the real and imaginary parts of $f(\boldsymbol{x})$ respectively, we propose a quantum algorithm that returns an $\varepsilon$-approximation of its gradient with query complexity $\widetilde{O}(1/\varepsilon)$. As an extension, we also propose two quantum algorithms for Hessian estimation, aiming to improve quantum analogs of Newton's method. The two algorithms have query complexity $\widetilde{O}(d/\varepsilon)$ and $\widetilde{O}(d^{1.5}/\varepsilon)$, respectively, under different assumptions. Moreover, if the Hessian is promised to be $s$-sparse, we then have two new quantum algorithms with query complexity $\widetilde{O}(s/\varepsilon)$ and $\widetilde{O}(sd/\varepsilon)$, respectively. We also prove a lower bound of $\widetildeΩ(d)$ for Hessian estimation in the general case.

Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.

With some mild assumptions on metric and topology of the central fiber, we prove that the limit of Kahler manifolds under holomorphic deformation is still Kahler.

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be transferred from one group to another through algebraic and geometric relations, such as quotients, subgroups, translation-like actions and quasi-isometries. We use the aforementioned tools to study the possible values taken by this invariant on subshifts of finite type on some finitely generated groups. We obtain a full classification for some classes, such as virtually polycyclic groups and branch groups with decidable word problem. We also show that all groups which are quasi-isometric to the hyperbolic plane admit SFTs with nonzero Medvedev degree. Furthermore, we provide a classification of the degrees of sofic subshifts for several classes of groups.

We show that irreducible strongly continuous representations of $\mathrm{SL}(2,\mathbb{R})$ on certain Banach spaces are admissible and that the admissibility of Banach space representations of SL(2,R) and the invariant subspace problem are intimately related.

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. The theorems are obtained in several settings, including holomorphic curves on $\mathbb{C}$, annuli, complex discs with finite growth index, and Kähler manifolds. We obtain a total defect bound that improves upon the previously known results. As an application, we establish a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

The abstract tile assembly model (aTam) is a model of DNA self-assembly. Most of the studies focus on cooperative aTAM where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings where assemblies do not need to cover the whole plane and some mismatches may occur). Introduced by P.E. Meunier, efficient paths are a non-trivial construction for non-cooperative aTAM designed with $n$ different tile types and reaching a distance linearly greater than n. Later, efficient paths were improved to be able to reach a distance of n log(n). Assembling them relies heavily on a form of ``non-determinism''. Indeed, the set of tiles may produce different finite terminal assemblies but they all contain the same efficient path, a model called directed non-cooperative aTAM. This variant of aTAM is the only one who was shown to be decidable. In this paper, we prove that this non-determinism is strictly necessary for assembling the efficient paths. This result also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.

We consider outbreak detection settings of endemic diseases where the population under study consists of various subpopulations available for stratified surveillance. These subpopulations can for example be based on age cohorts, but may also correspond to other subgroups of the population under study such as international travellers. Rather than sampling uniformly across the population, one may elevate the effectiveness of the detection methodology by optimally choosing a sampling subpopulation. We show (under some assumptions) the relative sampling efficiency between two subpopulations is inversely proportional to the ratio of their respective baseline disease risks. This implies one can increase sampling efficiency by sampling from the subpopulation with higher baseline disease risk. Our results require careful treatment of the power curves of exact binomial tests as a function of their sample size, which are non-monotonic due to the underlying discreteness. A case study of COVID-19 cases in the Netherlands illustrates our theoretical findings.

As the catalogue of gravitational-wave transients grows, several entries appear "exceptional" within the population. Tipping the scales with a total mass of $\approx 150 M_\odot$, GW190521 likely contained black holes in the pair-instability mass gap. The event GW190814, meanwhile, is unusual for its extreme mass ratio and the mass of its secondary component. A growing model-building industry has emerged to provide explanations for such exceptional events, and Bayesian model selection is frequently used to determine the most informative model. However, Bayesian methods can only take us so far. They provide no answer to the question: does our model provide an adequate explanation for the data? If none of the models we are testing provide an adequate explanation, then it is not enough to simply rank our existing models - we need new ones. In this paper, we introduce a method to answer this question with a frequentist $p$-value. We apply the method to different models that have been suggested to explain GW190521: hierarchical mergers in active galactic nuclei and globular clusters. We show that some (but not all) of these models provide adequate explanations for exceptionally massive events like GW190521.

The kinetic Sunyaev Zel'dovich (kSZ) effect is a blackbody cosmic microwave background (CMB) temperature anisotropy induced by Thomson scattering off free electrons in bulk motion with respect to the CMB rest frame. The statistically anisotropic cross-correlation between the CMB and galaxy surveys induced by the kSZ effect encodes the radial bulk velocity (more generally, the remote dipole field), which can be efficiently reconstructed using a quadratic estimator. Here, we develop a quadratic estimator for the remote dipole field for use with data from the Planck satellite and the unWISE galaxy redshift catalog. With this data combination, we forecast a signal-to-noise of order unity within $Λ$CDM assuming a simple model for the distribution of free electrons. Using reconstructions based on individual frequency temperature maps and a variety of component separated CMB maps, we characterize the impact of foregrounds and systematics. The dominant contaminant is a coupling between the cosmic infrared background and large-scale galaxy survey systematics. We develop a method to minimize this effect, and demonstrate that after doing so the reconstructions are consistent with the expected level and properties of reconstruction noise. We use this reconstruction to constrain the multiplicative optical depth bias characterizing the amplitude of the remote dipole field to $b_v < 1.04$ at $68 \%$ confidence. Our fiducial signal model with $b_v =1$ is consistent with this measurement. Our results support an optimistic future for kSZ velocity reconstruction with near-term datasets.

Strongly correlated materials feature multiple electronic orbitals which are crucial to accurately understand their many-body properties, from cuprate materials to twisted bilayer graphene. In such multi-band models, quantum interference can lead to dispersionless bands whose large degeneracy gives rise to itinerant magnetism even with weak interactions. Here, we report on signatures of a ferrimagnetic state realized in a Lieb lattice at half-filling, characterized by antialigned magnetic moments with antiferromagnetic correlations, concomitant with a finite spin polarization. We demonstrate their robustness when increasing repulsive interactions from the non-interacting to the Heisenberg regime, and study their emergence when continuously tuning the lattice unit cell from a square to a Lieb geometry. Our work paves the way towards exploring exotic phases in related multi-orbital models such as quantum spin liquids in kagome lattices and heavy fermion behavior in Kondo models.

The present study focuses on persistence in research productivity over the course of an individual's entire scientific career. We track 'late-career' scientists - scientists with at least 25 years of publishing experience (N=320,564) - in 16 STEMM (science, technology, engineering, mathematics, and medicine) and social science disciplines from 38 OECD countries for up to five decades. Our OECD sample includes 79.42% of late-career scientists globally. We examine the details of their mobility patterns as early-career, mid-career, and late-career scientists between decile-based productivity classes, from the bottom 10% to top 10% of the productivity distribution. Methodologically, we turn a large-scale bibliometric dataset (Scopus raw data) into a comprehensive, longitudinal data source for research on careers in science. The global science system is highly immobile: half of global top performers continue their careers as top performers and one-third of global bottom performers as bottom performers. Jumpers-Up and Droppers-Down are extremely rare in science. The chances of moving radically up or down in productivity classes are marginal (1% or less). Our regression analyses show that productivity classes are highly path dependent: there is a single most important predictor of being a top performer, which is being a top performer at an earlier career stage.

Rapid advances in the field of Large Language Models (LLMs) have made LLM-based code generation an important area for investigation. An LLM-based code generator takes a prompt as input and produces code that implements the requirements specified in the prompt. Many software requirements include mathematical formulas that specify the expected behavior of the code to be generated. Given a code generation prompt that contains a mathematical formula, a reasonable expectation is that, if the formula is syntactically modified without changing its semantics, the generated code for the modified prompt should be semantically equivalent. We formalize this concept as syntactic robustness and investigate the syntactic robustness of LLMs as code generators. Our experimental assessment demonstrates that LLMs are not syntactically robust for code generation prompts with formulas, especially for the ones that require mathematical reasoning. We investigate attack strategies that can further deteriorate the syntactic robustness of LLMs. Finally, to mitigate syntactic robustness failures in LLMs, we propose a pre-processing step that uses reductions to transform formulas in prompts to a simplified form. Our experimental results demonstrate that the syntactic robustness of LLM-based code generation improves significantly using our approach, improving syntactic robustness of LLMs from 54.05% to 74.42%.

We propose a method of spatial resolution enhancement in metrology (thermometry, magnetometry, pH estimation, and similar methods) with blinking fluorescent nanosensors by combining sensing with super-resolution optical fluctuation imaging (SOFI). By utilizing the idea of quantum super-resolution imaging by photon statistics (QSIPS), the applicability of the proposed methodology is extended to low-brightness regime. Efficiency of the approach is demonstrated by numerical simulations performed for several model configurations, representing step-like and continuous variation of the sensed parameter, high and low brightness regimes, 1- and 2-dimensional structures. The 2nd and 4th order cumulant images provide improvement of the contrast and enable successful reconstruction of smaller features of the modeled parameter distribution relatively to the intensity-based approach. We believe that blinking fluorescent sensing agents being complemented with the developed image analysis technique could be utilized routinely in the life science sector for recognizing the local changes in the spectral response of blinking fluorophores, e.g. delivered targetly to the specific cell or even organelle. It is extremely useful for the local measurements of living cells' physical parameters changes due to applying any external "forces", including disease effect, aging, healing, or response to the treatment.

National science systems have become embedded in global science and countries do everything they can to harness global knowledge to national economic needs. However, accessing and using the riches of global knowledge can occur only through scientists. Consequently, the research power of nations relies on the research power of individual scientists. Their capacity to collaborate internationally and to tap into the global networked science is key. The constantly evolving, bottom-up, autonomous, self-regulating, and self-focused nature of global science requires deeper understanding; and the best way to understand its dynamics is to understand what drives academic scientists in their work. The idea that science remains a state-driven rather than curiosity-driven is difficult to sustain. In empirical terms, we describe the globalization of science using selected publication, collaboration, and citation data from 2000-2020. The globalization of science implies two different processes in two different system types: the growth of science in the Western world is almost entirely attributable to internationally co-authored publications; its growth in the developing world, in contrast, is driven by both internationally co-authored and domestic publications. Global network science opens incredible opportunities to new arrivals - countries as well as institutions and research teams. The global system is embedded in the rules created by scientists themselves and maintained as a self-organizing system and nation-states have another major level to consider in their science policies: the global level. Globalization of science provides more agency, autonomy, collegiality, and self-regulation to scientists embedded in national science structures and involved in global networks.

We study the large-scale dynamics of charged particles in a rapidly oscillating field and formulate its classical and quantum effective theory description. The high-order perturbative results for the effective action are presented. Remarkably, the action models the effects of general relativity on the motion of nonrelativistic particles, with the values of the emergent curvature and speed of light determined by the field spatial distribution and frequency. Our results can be applied to a wide range of physical problems including the high-precision analysis and design of the charged particle traps and Floquet quantum materials.

Flipped classroom pedagogy is widely used in undergraduate mathematics to promote active learning, yet it remains unclear whether students experience it in systematically different ways. In this study, we analyze student perceptions from an introductory linear algebra course using survey data collected across multiple semesters. Using an interpretable machine learning approach, we examine patterns across questionnaire responses and evaluate their consistency under repeated analysis. Our results reveal a clear and stable separation in perception patterns when grouped by gender, suggesting that these differences arise from structured combinations of factors rather than isolated responses. The model also identifies key aspects of engagement and instructional design that contribute most to this separation. These findings highlight opportunities for more inclusive flipped classroom design and demonstrate the value of interpretable methods in educational research.

Spectral methods have myriad applications in high-dimensional statistics and data science, and while previous works have primarily focused on $\ell_2$ or $\ell_{2,\infty}$ eigenvector and singular vector perturbation theory, in many settings these analyses fall short of providing the fine-grained guarantees required for various inferential tasks. In this paper we study statistical inference for linear functions of eigenvectors and principal components with a particular emphasis on the setting where gaps between eigenvalues may be extremely small relative to the corresponding spiked eigenvalue, a regime which has been oft-neglected in the literature. First, we prove the approximate Gaussianity for debiased linear forms in the matrix denoising model and the spiked principal component analysis model, both under Gaussian noise. Based on this limiting behavior, we propose estimators for the appropriate bias and variance quantities resulting in approximately valid confidence intervals. We then investigate the optimality of these confidence intervals and show that their widths are minimax optimal up to constant factors. Of note, our proposed confidence intervals can be computed directly from data without the need for any sample-splitting.

The recently proposed orthogonal delay-Doppler division multiplexing (ODDM) modulation, which is a delay-Doppler (DD) domain multi-carrier (DDMC) modulation scheme based on the DD domain orthogonal pulse (DDOP), is studied. We first revisit the linear time-varying (LTV) channel model for the wireless channel, and review the conventional multi-carrier (MC) modulation schemes and their design guidelines for both linear time-invariant (LTI) and LTV channels. We then focus on the representation of the LTV channel in an equivalent sampled DD (ESDD) domain, and propose an impulse-function-based transmission strategy for the ESDD channel. Next, we take an in-depth look into the DDOP and show that it achieves orthogonality with respect to the fine time and frequency resolutions in the ESDD domain thus behaves like an impulse function. This allows us to unveil the unique input-output relation of the resultant ODDM modulation over the ESDD channel. We point out that the conventional MC modulation design guidelines based on the Weyl-Heisenberg (WH) frame theory can be relaxed without compromising its orthogonality or violating the WH frame theory. More specifically, for a practical communication system with bandwidth and duration constraints, MC modulation signals can be designed considering so-called local or sufficient (bi)orthogonality, which refers to the (bi)orthogonality among a WH subset for the MC signal within a specific bandwidth and duration. This novel design guideline could potentially open up opportunities for developing future waveforms required by new applications such as communication systems associated with high delay and/or Doppler shifts, as well as integrated sensing and communications.

A basic algorithm for enumerating disjoint propositional models (disjoint AllSAT) is based on adding blocking clauses incrementally, ruling out previously found models. On the one hand, blocking clauses have the potential to reduce the number of generated models exponentially, as they can handle partial models. On the other hand, the introduction of a large number of blocking clauses affects memory consumption and drastically slows down unit propagation. We propose a new approach that allows for enumerating disjoint partial models with no need for blocking clauses by integrating: Conflict-Driven Clause-Learning (CDCL), Chronological Backtracking (CB), and methods for shrinking models (Implicant Shrinking). Experiments clearly show the benefits of our novel approach.

In this paper we prove that a totally integrable strictly-convex symplectic billiard table, whose boundary has everywhere strictly positive curvature, must be an ellipse. The proof, inspired by the analogous result of Bialy for Birkhoff billiards, uses the affine equivariance of the symplectic billiard map.

Cursed Equilibrium of Eyster and Rabin (2005) has been a leading theory for explaining winner's-curse-type behavior in static Bayesian games, but it faces conceptual limitations when applied to dynamic games. Two recent extensions, Cursed Sequential Equilibrium (CSE) by Fong, Lin and Palfrey (2025) and Sequential Cursed Equilibrium (SCE) by Cohen and Li (2026), address these limitations in fundamentally different ways. Complementing these two papers, this paper provides a systematic comparison of CSE and SCE, clarifying their conceptual foundations and technical implications, including their notions of cursedness, belief updating, and treatment of public histories.