"Addition-by-subtraction" coupled cluster (CC) approaches provide a promising approach to treating the difficult strong correlation problem by simplifying the standard CC equations. In a separate vein, linearized CC methods have drawn interest for their lower computational cost, increased parallelizability, and favorable properties for extension to the excited state--but the inclusion of ring/crossed-ring terms causes singularities even for single bond breaking. A linearized, addition-by-subtraction CC method called linearized ladder CCD (linLCCD) removes these terms to avoid divergences, but linLCCD under-estimates dynamical correlation. Herein we resolve this deficiency of linLCCD by introducing a linearized external coupled cluster perturbation theory that adds a second-order ring/crossed-ring correction back into a linLCCD reference wave function. Our resultant xlinCCD(2) method is regular and yields comparable results to linearized CCD in weakly-correlated regimes.

Personal computers and handheld devices provide keyboard shortcuts and swipe gestures to enable users to efficiently switch between applications, whereas today's virtual reality (VR) systems do not. In this work, we present an exploratory study on user interface aspects to support efficient switching between worlds in VR. We created eight interfaces that afford previewing and selecting from the available virtual worlds, including methods using portals and worlds-in-miniature (WiMs). To evaluate these methods, we conducted a controlled within-subjects empirical experiment (N=22) where participants frequently transitioned between six different environments to complete an object collection task. Our quantitative and qualitative results show that WiMs supported rapid acquisition of high-level spatial information while searching and were deemed most efficient by participants while portals provided fast pre-orientation. Finally, we present insights into the applicability, usability, and effectiveness of the VR world switching methods we explored, and provide recommendations for their application and future context/world switching techniques and interfaces.

The vortex-induced vibration of multiple spring-mounted bodies free to move in the orthogonal direction of the flow is investigated. In a first step, we derive a Linear Arbitrary Lagrangian Eulerian (L-ALE) method to solve the fluid-structure linear problem as well as a forced problem where a harmonic motion of the bodies is imposed. We then propose a low computational-cost impedance-based criterion to predict the instability thresholds. A global stability analysis of the fluid-structure system is then performed for a tandem of cylinders and the instability thresholds obtained are found to be in perfect agreement with the predictions of the impedance-based criterion. An extensive parametric study is then performed for a tandem of cylinders and the effects of mass, damping and spacing between the bodies are investigated. Finally we also apply the impedance-based method to a three-body system to show its validity to a higher number of bodies.

In this paper, we characterize the extreme points of a class of multidimensional monotone functions. This result is then applied to large contests, where it provides a useful representation of optimal allocation rules under a broad class of distributional preferences of the contest designer. In contests with complete information, the representation significantly simplifies the characterization of the equilibria.

We study the kinematics of condensations in 30 fields forming high-mass stars with ALMA at a high-resolution of ~0.08'' on average (~230 au). The presence of disks is important for feeding high-mass stars without feedback halting growth as their masses increase. In the search for velocity gradients resembling rotation that can reveal the presence of disks, we analyze the emission of gas tracers in 49 objects using CH$_3$OH, CH$_3$CN, and tentative detections of HNCO and cis-HCOOH. Most of the velocity distributions show velocity gradients indicative of rotation. We reveal a total of 32 disk candidates, the largest sample to date that has been uniformly analyzed at a few hundred au scales in the high-mass regime. Their position-velocity maps are generally asymmetric with one side brighter than the opposite. We successfully fit a power law to the position-velocity maps of the disk candidates and find indices between -0.5 (Keplerian rotation) and -1 (rotation under specific angular momentum conservation) with a median of -0.7. Under Keplerian rotation assumption, we estimate central masses, uncorrected for inclination, ranging between 7 to 45 M$_\odot$. Excluding outliers, the disk candidates are relatively more compact (<200 au) and less massive (<5 M$_\odot$) than previous results at coarser angular resolution. We calculate an average Toomre-$Q$ parameter and find that most are gravitationally unstable (median of 0.5). We conclude that these observations offer the first opportunity to separate the disk and envelope components of hot cores on a statistically significant sample, and confirm that anisotropic collapse plays an role in feeding high-mass (proto)stars.

Why do division and cohesion often intensify together? This paper develops a static structural theory of relation maintenance based on minimal positional constraints. Rather than relying on utility-based or probabilistic models, social relations are formalized as constraint satisfaction problems over an abstract position space. When a bifurcation event -- such as a vote or institutional assignment -- fixes agents' positions, relational viability is determined solely by positional compatibility. We establish three structural results. First, under any non-degenerate positional constraint, fragmentation (relational collapse) and cohesion (condition confirmation) necessarily coexist as complementary outputs of a single compatibility function. Second, we prove a structural asymmetry of veto power: relation maintenance requires logical conjunction, while collapse requires only logical disjunction, implying that fragmentation operates as a unilateral structural veto. This yields a purely logical foundation for behavioral premises such as pairwise stability. Finally, we establish a conditional impossibility theorem: under positional plurality, avoiding relational collapse is structurally impossible, leaving coercive homogenization as the only design-level guarantee for universal cohesion. The framework isolates minimal boundary conditions and provides a formal language for analyzing polarized relational structures.

Retrieval is being redefined by agentic AI, demanding multimodal reasoning beyond conventional similarity-based paradigms. Composed Image Retrieval (CIR) exemplifies this shift as each query combines a reference image with textual modifications, requiring compositional understanding across modalities. While embedding-based CIR methods have achieved progress, they remain narrow in perspective, capturing limited cross-modal cues and lacking semantic reasoning. To address these limitations, we introduce XR, a training-free multi-agent framework that reframes retrieval as a progressively coordinated reasoning process. It orchestrates three specialized types of agents: imagination agents synthesize target representations through cross-modal generation, similarity agents perform coarse filtering via hybrid matching, and question agents verify factual consistency through targeted reasoning for fine filtering. Through progressive multi-agent coordination, XR iteratively refines retrieval to meet both semantic and visual query constraints, achieving up to a 38% gain over strong training-free and training-based baselines on FashionIQ, CIRR, and CIRCO, while ablations show each agent is essential. Code is available: https://01yzzyu.github.io/xr.github.io/.

The Korteweg-de Vries (KdV) equation governs the propagation of nonlinear internal and surface gravity waves in shallow ocean environments, where the balance between nonlinear steepening and frequency-dependent dispersion produces solitons. This article presents sangkuriang, an open-source Python library that solves the KdV equation using Fourier pseudo-spectral spatial discretization and adaptive eighth-order Runge-Kutta time integration, accelerated via just-in-time (JIT) compilation. Validation across four progressively complex scenarios-isolated soliton propagation, symmetric interactions, overtaking collisions, and three-body interactions-demonstrates high-fidelity conservation of mass, momentum, and energy, with relative errors below $O(10^{-4})$. Measured soliton velocities agree with theoretical predictions within $5\%$, and complementary diagnostics based on spectral entropy and recurrence quantification analysis (RQA) confirm that computed solutions preserve the regular phase-space structure characteristic of integrable Hamiltonian systems. Running on a standard laptop, sangkuriang provides a robust, lightweight platform for reproducible numerical investigation of idealized nonlinear dispersive wave dynamics relevant to coastal and ocean engineering applications.

Neuromorphic computers hold the potential to vastly improve the speed and efficiency of a wide range of computational kernels with their asynchronous, compute-memory co-located, spatially distributed, and scalable nature. However, performance models that are simple yet sufficiently expressive to predict runtime on actual neuromorphic hardware are lacking, posing a challenge for researchers and developers who strive to design fast algorithms and kernels. As breaking the memory bandwidth wall of conventional von-Neumann architectures is a primary neuromorphic advantage, modeling communication time is especially important. At the same time, modeling communication time is difficult, as complex congestion patterns arise in a heavily-loaded Network-on-Chip. In this work, we introduce the first max-affine lower-bound runtime model -- a multi-dimensional roofline model -- for Intel's Loihi 2 neuromorphic chip that quantitatively accounts for both compute and communication based on a suite of microbenchmarks. Despite being a lower-bound model, we observe a tight correspondence (Pearson correlation coefficient greater than or equal to 0.97) between our model's estimated runtime and the measured runtime on Loihi 2 for a neural network linear layer, i.e., matrix-vector multiplication, and for an example application, a Quadratic Unconstrained Binary Optimization solver. Furthermore, we derive analytical expressions for communication-bottlenecked runtime to study scalability of the linear layer, revealing an area-runtime tradeoff for different spatial workload configurations with linear to superlinear runtime scaling in layer size with a variety of constant factors. Our max-affine runtime model helps empower the design of high-speed algorithms and kernels for Loihi 2.

Formation of a rapidly spinning, strongly magnetized neutron star (NS) may occur in various classes of core-collapse events. If the NS injects an amount of energy comparable to the explosion energy of the accompanying supernova (SN) before the SN ejecta becomes transparent, the nascent NS wind bubble can overtake the outer ejecta and undergo a blowout driven by hydrodynamic instabilities. Based on multidimensional numerical studies, we construct a minimal semi-analytic framework to follow the post-blowout dynamics and radiative evolution, map the blowout conditions by scanning the ejecta and NS parameters, and compute survey-ready multi-band light curves. For stripped-envelope SNe with an ejecta mass of $M_\mathrm{ej} \sim 10\,M_\odot$ and an explosion energy of $E_\mathrm{sn} \sim 10^{51}\,\mathrm{erg}$, blowout occurs for NSs with magnetic field strengths of $B_{\mathrm{dip}} \gtrsim 10^{13}\,\mathrm{G}$ and spin periods of $P_\mathrm{NS} \lesssim \mathrm{a\ few}\,\mathrm{ms}$. Relatively weak-field cases with $B_\mathrm{dip} \lesssim 10^{14}\,\mathrm{G}$ produce luminous double-peaked UV/optical light curves, as observed in the superluminous SN LSQ14bdq, while stronger-field cases with $B_\mathrm{dip} \gtrsim 10^{14}\,\mathrm{G}$ result in hypernovae preceded by X-ray blowout precursors. We also examine weaker and lower-mass SN explosions representing ultra-stripped SNe and accretion- or merger-induced collapse events, in which blowout is more readily achieved over a broader range of NS parameters, producing fast X-ray transients with durations of $ 10^{2\mbox{--}4}\,\mathrm{s}$ and peak luminosities of $10^{42\mbox{--}48}\,\mathrm{erg\,s^{-1}}$. Our results encourage coordinated UV, optical, and X-ray observations which constrain the formation of the most energetic NSs in the universe.

In the study of factorizations of finite cyclic groups, a classical problem is to investigate the properties of factorization sets $A$ and $B$ in the direct sum decomposition $A \oplus B = \mathbb{Z}_{M}$ with $|A| = |B| =\sqrt{M}$, where $M=(pqr)^2$ for some distinct primes $p$, $q$, and $r$. In this paper, we show that neither $A$ nor $B$ is contained in a proper subgroup of $\mathbb{Z}_{(pqr)^2}$ if and only if the factorization sets $A, B$ form a Szabó pair. The factorization of finite cyclic groups is closely connected to the properties of tiling and spectral sets in $\Bbb Z$. The problem considered in this paper is equivalent to the simplest form of tiling that cannot be reduced to the two--prime case by the method provided by Coven and Meyerowitz (J. Algebra 212: 161--174, 1999). In contrast, the construction for the tiling which can be reduced to the two--prime case is already known. Our results present full structures for the factorization sets $A$ and $B$, and therefore, for this class of tilings.

Recent experiments on few-layer graphene structures have reported indirect signatures of anomalous Hall crystals (AHCs), but the need for a top gate to stabilize the phase precludes direct imaging of the emergent electronic lattice. This situation necessitates the investigation of alternative signatures of AHCs. The gapless phonons of the emergent electronic lattice provide a clear distinction from conventional quantum Hall states, but it may be difficult to disentangle these phonons from the plethora of other possible low-lying modes. Intriguingly, the quantum geometry of the underlying electronic ground state can imprint on the collective modes, possibly leading the phonons themselves to be topological. Were this the case, the resulting neutral chiral edge modes would provide a further signature of an AHC. Using time-dependent Hartree-Fock, we compute the spectra of collective modes of Wigner crystals (WCs) and AHCs arising in minimal models and study the topology of the phonons and low-lying excitons. Across the WC to AHC transition, we observe a series of band inversions among collective modes, producing topological phonons and excitons, and a sharp sign change in the phonon Chern number upon entering the AHC phase. We conclude by discussing the relevance of collective mode topology to experiments on candidate systems for AHCs.

This paper considers estimation and model selection of quantile vector autoregression (QVAR). Conventional quantile regression often yields undesirable crossing quantile curves, violating the monotonicity of quantiles. To address this issue, we propose a simplex quantile vector autoregression (SQVAR) framework, which transforms the autoregressive (AR) structure of the original QVAR model into a simplex, ensuring that the estimated quantile curves remain monotonic across all quantile levels. In addition, we impose the smoothly clipped absolute deviation (SCAD) penalty on the SQVAR model to mitigate the explosive nature of the parameter space. We further develop a Bayesian information criterion (BIC)-based procedure for selecting the optimal penalty parameter and introduce new frameworks for impulse response analysis of QVAR models. Finally, we establish asymptotic properties of the proposed method, including the convergence rate and asymptotic normality of the estimator, the consistency of AR order selection, and the validity of the BIC-based penalty selection. For illustration, we apply the proposed method to U.S. financial market data, highlighting the usefulness of our SQVAR method.

We investigate the combined effects of spatial curvature and topology on the properties of the vacuum state for a charged scalar field localized on the (2+1)-dimensional Beltrami pseudosphere, assuming that the field obeys quasiperiodicity condition with constant phase. As important local characteristics of the vacuum state the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor are evaluated. The contributions in the VEVs coming from geometry with an uncompactified azimuthal coordinate are divergent, whereas the compact counterparts are finite and are analysed both numerically and asymptotically. For small values of proper radius of the compactified dimension, the leading terms of topological contributions are independent of the field mass and curvature coupling parameter, increasing by a power-law. In the opposite limit, the VEVs decay following a power-law in the general case. In the special case of a conformally coupled massless field the behavior is different. Unlike the VEV of field squared and vacuum energy density, the radial and azimuthal stresses are increasing by absolute value. As a consequence, the effects of nontrivial topology are strong for the stresses in this case at small values of radial coordinate.

Regardless of their initial spins, the merger of two roughly equal mass black holes (BHs) produces a remnant BH of dimensionless spin $0.69$. Such remnants can merge with other BHs in dense stellar environments and produce hierarchical mergers. Analyzing the latest catalog binary black hole (BBH) mergers from the LIGO-Virgo-KAGRA detectors, we identify a subpopulation with primary spins consistent with such hierarchical mergers. Consistent with astrophysical expectations for mergers of second-generation BHs with first-generation BHs, we find that this subpopulation has mass ratios below $0.59^{+0.18}_{-0.23}$. We also infer that $19$-$88\%$ of the BBH population below this mass ratio is consistent with belonging to the hierarchically merged population. Our results offer a natural explanation for the narrowing of the effective inspiral spin distribution with mass ratio observed in other studies.

This paper presents a method to identify electrons using the Cherenkov light emitted when a charged particle travels in air and photons are detected with a Silicon PhotoMultiplier (SiPM). The analysis is based on a photon-counting approach using SPAD cells and uses data collected during a test beam at CERN PS. The results are well described by a simple Monte Carlo simulation, which further demonstrates that a very good electron identification and a strong pion/hadron rejection could be obtained over a wide momentum range.

Insufficient link budget has become a bottleneck problem for direct access in current satellite communications. In this paper, we develop a semantic transmission framework for direct satellite communications as an effective and viable solution to tackle this problem. To measure the tradeoffs between communication, computation, and generation quality, we introduce a semantic efficiency metric with optimized weights. The optimization aims to maximize the average semantic efficiency metric by jointly optimizing transmission mode selection, satellite-user association, ISL task migration, denoising steps, and adaptive weights, which is a complex nonlinear integer programming problem. To maximize the average semantic efficiency metric, we propose a decision-assisted REINFORCE++ algorithm that utilizes feasibility-aware action space and a critic-free stabilized policy update. Numerical results show that the proposed algorithm achieves higher semantic efficiency than baselines.

Divisor methods are well known to satisfy house monotonicity, which allows representative seats to be allocated sequentially. We focus on stationary divisor methods defined by a rounding cutpoint $c \in [0,1]$. For such methods with integer-valued votes, the resulting apportionment sequences are periodic. Restricting attention to two-party allocations, we characterize the set of possible sequences and establish a connection between the lexicographical ordering of these sequences and the parameter $c$. We then show how sequences for all pairs of parties can be systematically extended to the $n$-party setting. Further, we determine the number of distinct sequences in the $n$-party problem for all $c$. Our approach offers a refined perspective on size bias: rather than viewing large parties as simply receiving more seats, we show that they instead obtain their seats earlier in the apportionment sequence. Of particular interest is a new relationship we uncover between the sequences generated by the smallest divisor (Adams) and greatest divisor (D'Hondt or Jefferson) methods.

Topologically constrained genome-like polymers often double-fold into tree-like configurations. Here we calculate the exact number of tightly double-folded configurations available to a ring polymer in ideal conditions. For this purpose, we introduce a scheme which allows us to define a ``code'' specifying how a ring wraps a randomly branching tree and calculate the number of admissible wrapping codes via a variant of Bertrand's ballot theorem. As a validation, we demonstrate that data from Monte Carlo simulations of an elastic lattice model of non-interacting tightly double-folded rings with controlled branching activity are in excellent agreement with exact expressions for branch-node and tree size statistics that can be derived from our expression for the ring entropy.

Contemporary AI regulation, including the EU Artificial Intelligence Act and related governance frameworks, increasingly requires institutions to justify the training data used in automated decision-making. Yet existing governance regimes provide limited operational methods for selecting, weighting, and explaining data inputs. We introduce the Smart Data Portfolio (SDP) framework, which treats data categories as productive but risk-bearing assets, formalizing input governance as an information-risk trade-off. Within this framework, we define two portfolio-level quantities, Informational Return and Governance-Adjusted Risk, whose interaction characterizes attainable data mixtures and yields a Governance-Efficient Frontier. Regulators shape this frontier through risk caps, admissible categories, and weight bands that translate fairness, privacy, robustness, and provenance requirements into measurable constraints on data allocation while preserving model flexibility. A sectoral illustration shows how different AI services require distinct portfolios within a common governance structure. The framework provides an input-level explanation layer through which institutions can justify governed data use in large-scale AI deployment.

Simulating one-dimensional stellar evolution models with MESA, we show that removing the outer inflated envelope of a mass-accreting evolved stripped-envelope star, like a Wolf-Rayet (WR) star, substantially moderates the stellar expansion during accretion at high-mass accretion rates. We study the accretion onto a star via an accretion disk, which launches jets that remove the high-entropy outer layers of the inflated envelope. This is the `jetted mass removal accretion scenario.' By manually removing the entire hydrogen-rich envelope from a red supergiant, we build a hydrogen-deficient WR stellar model with a mass of 6.03Mo and a radius of 0.67Ro. We then accrete mass onto it at a high rate. We mimic the real process of simultaneous mass addition near the equatorial plane and jet-induced mass removal from the outer envelope by dividing the accretion period into hundreds of pulses: in the first half of each pulse, we add mass, and in the second, we remove a fraction of this mass. The removal of tens of percent from the added mass decreases the stellar expansion by a factor of ~2-5. Our results show that WR stars can maintain a deep potential well and not expand much while accreting mass at high rates. This allows the formation of an accretion disk and the liberation of large amounts of gravitational energy. Our results strengthen models of intermediate-luminosity optical transients, such as luminous red novae, in which a non-degenerate star accretes at high rates and launches jets that power the transient event.

Copulas are the primary tool for dependence modeling in statistics, and quasi-copulas are their essential companions. The latter appear, say, as infima or suprema of sets of copulas; they form a huge class and have some unpleasant properties. Their statistical interpretation is challenged by the fact that they may lead to negative volumes of some boxes. So, numerous applications call for an intermediate class, and supermodular quasi-copulas are one of them, having many useful properties. An excellent measure, Average Rectangular Volume (ARV in short), to clarify and position this class was proposed in the seminal paper by Anzilli and Durante, The average rectangular volume induced by supermodular aggregation functions, J. Math. Anal. Appl. 555 (2026) 21 pp. While supermodularity is a bivariate notion, its extension to the $d$-variate case for $d>2$ was recently emphasized in a key paper by Arias-Garcia, Mesiar, and De Baets, The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions, Int. J. of Appr. Reasoning, 80 (2017) pp. 89-99. Here, an alternative method to ARV is presented, extendable to the multivariate case based on Maximal (in absolute value) Negative Volumes (MNV in short) on boxes, thus helping practitioners when seeking the right (quasi-)copula for their problem. Observe that these volumes on copulas are zero, while their values on quasi-copulas, depending on $d$, have been a long-standing open problem solved only recently. We present a nontrivial extension of this solution, which serves as the main goal of this paper: a measure that clarifies and positions the classes considered based on MNV.

Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly high-dimensional, dynamical systems is far from straightforward, as computational methods have to meet a delicate trade-off involving stability, accuracy and computational cost. In this paper we investigate the design, and theoretical features, of constrained Bayesian filters for state space models. The constraint on the filter is given by a sequence of compact subsets of the state space that determines the sources and targets of the Markov transition kernels in the dynamical model. Subject to such constraints, we provide sufficient conditions for filter stability and approximation error rates with respect to the original (unconstrained) Bayesian filter. Then, we look specifically into the implementation of constrained filters in a continuous-discrete setting where the state of the system is a continuous-time stochastic Itô process but data are collected sequentially over a time grid. We propose an implementation of the constraint that relies on a data-driven modification of the drift of the Itô process using barrier functions, and discuss the relation of this scheme with methods based on the Doob $h$-transform. Finally, we illustrate the theoretical results and the performance of the proposed methods in computer experiments for a partially-observed stochastic Lorenz 96 model.

The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T.~(\textit{Math.~Ann.}~2024) regarding the existence of large subspaces of $C(K)$ that perform stable phase retrieval (SPR). We begin by proving that for both the real and complex fields, the space $C(K)$ admits an infinite-dimensional SPR subspace if and only if the second Cantor-Bendixson derivative $K{''}$ is nonempty. We then show how to construct ``large" SPR subspaces of $C(K)$, where the size of the subspace depends quantitatively on the number of non-trivial Cantor-Bendixson derivatives that the compact Hausdorff space $K$ possesses.

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(α)}\neq\varnothing$, where $2<α<ω$, then $C[1,ω^α]$ isometrically embeds into $C(K)$ doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case $α=2$.

Field space geometry plays a central role within the Swampland Programme, most notably in the various Distance Conjectures. However, for gravitational EFTs, this geometry is not uniquely defined: one can cast the action in many synonymous descriptions related by Weyl transformations, in which the field space metric transforms non-trivially across conformal frames. This raises a crucial question of how we are meant to think of the field space metric in view of employing the Swampland Conjectures. In this work we resolve this ambiguity by developing a fully frame-covariant framework for studying gravitational EFTs. We show that all conformal frames arise as distinct foliations of a singular higher-dimensional auxiliary geometry. Applying ADM formalism to the augmented field space, it is clear how Weyl- and unit transformations can be understood from a geometric point of view. Using this framework, we revisit the Species Scale Distance Conjecture and Sharpened Distance Conjecture, and show how the bounds derive from universal properties of gravitational EFTs under Weyl transformations. This strongly suggests that aspects of these conjectures apply to a much broader class of scalar-tensor theories and are consequences of frame covariance, rather than constraints imposed by quantum gravity.

In this article, we generalize some results in Chan-Yuan [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 26 (2025), 619--644] to local holomorphic maps between Hermitian manifolds preserving $(p,p)$-forms. In particular, we obtain further rigidity theorems and non-existence theorems for such maps.

In the context of complex algebraic varieties, the decomposition theorem for semi-small maps provides a decomposition of the direct image of the constant sheaf. In this work, we develop a decomposition theorem for branched coverings of topological spaces. To achieve this, we start by constructing a decomposition theorem for unramified covering maps via an explicit gluing construction using transition functions. For a given branched covering of closed topological manifolds, we use the previous result to establish a decomposition of the direct image of the constant sheaf on the covering space. In the next step, we generalize our discussion to the case where the target space is not necessarily a topological manifold.

We observe anomalous charge modulations induced by ionized indium impurities on the surface of the semiconductor black phosphorus by scanning tunneling microscopy (STM). When the impurities are switched into a negatively charged state by the STM tip, periodic charge modulations emerge around the impurity center, but strictly confined by the nanoscale impurity potential. These modulations form a distorted triangular pattern, whose periodicity remains unchanged in a wide range of positive bias. Furthermore, these local charge orders exhibit an anisotropy opposite to that expected based on the anisotropy of the Fermi surface, challenging a simple band-structure interpretation. Our experiment demonstrates the possibility of creating and manipulating macroscopic charge orders through impurity engineering.

This paper consists of three parts: (a) exhibit a new gluing result which can dramatically simplify extensions of calibration pairs; (b) observe that every Lawlor cone can support coflat calibrations singular only at the origin; (c) show that there exist many Lawlor cones which cannot support any smooth calibrations. As an application, we extend our previous work on detecting duality obstruction of calibrations in the smooth category.