AI for Science

The Haldane-Shastry spin chain is a prototypical integrable model with long-range interactions, notable for hosting quasiparticles with fractional statistics and serving as a discrete analogue of a conformal field theory. Its remarkable simplicity is closely tied to a full Yangian spin symmetry. While the highest-weight states for this symmetry are known explicitly, a systematic treatment of the descendant states, needed for the computation of various physical quantities, has remained incomplete. In this work, we provide a detailed construction of these descendants in terms of the algebraic Bethe ansatz following recent work of Ferrando et al. In the limit of extreme twist, it includes the Gelfand-Tsetlin basis. As an application, we derive product and determinant formulae for norms and overlaps of these states.

Bacteria often swim in complex environments where surfaces are ubiquitous and rarely flat. Surface topography and curvature can strongly affect bacterial motility, with important consequences for surface exploration, adhesion, and biofilm formation. Here, we investigate the swimming of a non-tumbling \textit{Escherichia coli} bacterium near an undulating no-slip surface using hydrodynamic simulations of a detailed model bacterium. The latter is described by a rigid spherocylindrical cell body and flexible flagella modeled with the Kirchhoff rod theory, while the surrounding fluid is simulated using the method of multi-particle collision dynamics. At low curvatures of the sinusoidal surface modulations, the bacterium exhibits persistent near-surface swimming and clockwise trajectories, consistent with the known behavior near flat no-slip walls. As the curvature increases, bacteria swimming toward a ridge can escape from the surface, which we use to estimate a critical curvature where surface detachment is more likely. At larger curvatures, we find that the surface geometry promotes oscillatory swimming along the groove direction, which reduces escape opportunities and, therefore, enhances bacterial trapping. Indeed, the confinement around the groove reverses the swimming of the bacterium from clockwise to counter-clockwise, as we demonstrate by two minimal models. Thus our work highlights the importance of the three-dimensional surface topography in bacterial surface exploration.

We formulate a thermodynamic model of a nano-reactor containing charge-regulated macroions within an electrolyte-permeable enclosure. The model is then formalized within the Poisson-Boltzmann electrostatics augmented by the consistent inclusion of the charge dissociation of molecular groups residing on the surface of the entrapped macroions via charge regulation formalism. By solving the basic equilibrium equations in the linearized Debye-Hückel type approximation, we analyze the salient features of the inhomogeneous electrolyte distribution and macroion charge. We found that the surface charge asymmetry/symmetry of the macroions strongly affects the spatial profile of electrostatic potential. The effective screening length shows the non-monotonic behavior, arising from the complex interplay between the bathing external solution and macroion effective charges, which govern charge regulation equilibria. The total pressure at the nano-reactor enclosure boundary decreases monotonically as the enclosure radius and the ionic bulk salt concentration increase. Also, the resulting pressure is strongly influenced by the surface charge densities of the nano-reactor and the number of confined macroions.

The scanning tunneling microscope (STM) is a powerful tool for investigating and manipulating molecules on surfaces. We demonstrate with a low-temperature STM operated at 6.2 K the controlled manipulation of ternary clusters of persistent molecular $π$ radicals as a whole. The ternary clusters $-$ each self-assembled from three $α,γ$-bisdiphenylene-$β$-phenylallyl (BDPA) molecules on Au(111) $-$ maintain their natural cluster structure throughout tip-induced translation and rotation relative to the surface. Sustained and repeated dragging of radical clusters is shown to facilitate the construction of artificial assemblies of several clusters. Our results provide new opportunities for the creation and investigation of radical-based spin assemblies on surfaces.

We consider the three-dimensional Bose polaron problem in the regime of finite range interactions and competing length scales. Working in the reference frame of the impurity, we study both static and out of equilibrium properties of the system, in particular the transfer of momentum between the impurity and the host gas. We find that relaxation dynamics can occur via damped oscillations of the impurity velocity with simple dependence on the interaction strength. Furthermore, the equilibration process is sensitive to the type of the impurity-bath interaction. Specifically, interatomic forces describing ion-atom systems lead to much longer timescales and more pronounced oscillations in the strong coupling regime with respect to local interaction potentials. We also find that the effective masses can differ by a large amount between the two scenarios, even if the number of atoms in the polaron cloud remains similar for both cases.

Recent experiments combining electrophoresis with pressure-driven flows in microchannels have revealed that microparticles undergo lateral migration perpendicular to the applied electric field. Although fluid inertia has been proposed as a possible explanation, inertial effects are negligibly small in these regimes, leaving the underlying physical mechanism an open question. In this study, we address these observations by extending previous theoretical work on concentration polarization,i.e., the external-field-induced modification of the ionic concentration field surrounding a dielectric object. We consider a dielectric particle with surface conductance subjected simultaneously to an external electric field and a shear flow. We show that the shear flow breaks the symmetry of the ionic concentration around the particle in the direction perpendicular to the applied field, thereby driving lateral migration. We demonstrate that the resulting migration velocity comprises two distinct contributions: an electrophoretic and a diffusiophoretic component. Our theory yields an explicit expression for the velocity magnitude as a function of the zeta potential and the Dukhin number, predicting typical speeds on the order of $\mathrmμ$m/s for representative experimental parameters. Notably, the model also predicts a reversal in the migration direction for Dukhin numbers of order unity.

We introduce a shape-based polar order parameter that captures the structural asymmetry of cells within epithelial monolayers. By combining bright-field imaging and traction force microscopy, we demonstrate that shape polarity serves as a unifying biomechanical metric, integrating the physical information encoded by nematic directors, principal stresses, and cellular motion. Furthermore, we show that the tissue organizes into a mixed polar-nematic phase, characterized by the coexistence of integer ($\pm 1$) and half-integer ($\pm 1/2$) defects. Through mechanical perturbations, we demonstrate that both substrate stiffness and cell-cell adhesion modulate the density of these excitations and the length of domain walls binding like-signed positive half-integer defects. Using a minimal continuum model of polar-nematic active matter, we establish that this mixed phase is fundamentally driven by the interplay of active stresses and polar-nematic elasticity. These findings provide a direct experimental evidence that epithelial monolayers behave as nematopolar matter, in which coupled polar and nematic elastic interactions jointly shape the active state

One way to construct Difference Distance Magic Oriented Graphs (DDMOGs) is via a recently introduced technique called weighted sum. We explore the quality of said construction further by introducing the notion of an ornament. An ornament is an oriented graph that, when used in a weighted sum with an existing DDMOG, creates a new DDMOG. We provide results on the construction of a specific type of ornaments, called s-nodes ornaments. We conclude the paper with a list of open questions related to DDMOGs and ornaments.

We construct the Higgs and Yukawa sectors as a structural completion of an algebraic three-generation model based on the complex Clifford algebra $\mathbb{C}\ell(10)$ with an intrinsic $S_3$ family symmetry. This addresses a common limitation of algebraic frameworks, in which Standard Model fermion multiplets and gauge symmetries may be described naturally, while the Higgs and Yukawa sectors remain less developed or absent. In the present framework, three algebraically distinguished fermion sectors are permuted by $S_3$, while the Standard Model gauge generators remain generation-independent. Higgs components are realised as right-action operators mapping weak-doublet fermion sectors into the corresponding weak-singlet sectors, and Yukawa coefficients are extracted using a Hilbert--Schmidt trace pairing. This yields two first-generation Higgs doublets with electroweak quantum numbers $(1,2,-1)$ and $(1,2,+1)$ under $SU(3)_C \times SU(2)_L \times U(1)_Y$, together with a Type-II-like separation between down-type and up-type Yukawa channels. Acting with the order-three family generator then generates a family-resolved Higgs sector organised into cyclic $S_3$ orbits. In the cyclically averaged Higgs limit, the Type-II-like Yukawa selection rule is preserved, while the generation-space Yukawa matrix is fixed algebraically and is non-diagonal in the algebraic generation basis. Under the usual implementation of electroweak symmetry breaking, the neutral Higgs couplings are aligned with the corresponding mass matrices, so tree-level flavour-changing neutral currents are not expected in this limit. The result is a constrained algebraic starting point for future $S_3$-breaking flavour phenomenology.

Let $β$ be a non-unit real algebraic integer greater than one and $\{a_{n}\}_{n \geq 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=aa_{n+2}+ba_{n+1}+ca_{n}$. Under certain conditions, we prove that the number of $a_{n}$ which are palindromic concatenations of two repdigits in base $β$ is finite.

We investigate Diophantine pairs and triples of triangular numbers with the property $D(a)$ for a non-zero integer $a$. We prove that if a triangular number belongs to a $D(a)$-pair, it can be extended to infinitely many $D(a)$-triples of triangular numbers. Additionally, we determine infinite families of integers $a$ that admit such pairs, as well as families for which no $D(a)$-pairs can exist.

Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an equivalence relation called cut-concordance, which encompasses the topological notion of concordance for classical links. Our main result is that the nilpotent peripheral system of 1-dimensional cut-diagrams is an invariant of cut-concordance, giving along the way a combinatorial version of a theorem of Stallings. We also investigate the relationship with several other equivalence relations in diagrammatic knot theory, in particular in connection with link-homotopy.

Folkman's theorem asserts the existence of graphs $G$ which are $K_4$-free, but which have the property that every two-coloring of $E(G)$ contains a monochromatic triangle. The quantitative aspects of $f(2,3,4)$, the least $n$ such that there exists an $n$-vertex graph with both properties above, are notoriously difficult; a series of improvements over the span of two decades witnessed the solution to two \$100 Erdős problems, and the current record due to Lange, Radziszowski, and Xu now stands at $f(2,3,4) \leq 786$,with another \$100 problem of Graham asking for a proof that $f(2,3,4) < 100$. In this paper, we study Folkman-like properties of a sequence $H_q$ of finite geometric graphs constructed using Hermitian unitals in projective planes and present some evidence that the graph $H_3$, which has 63 vertices, might contain a Folkman graph as a proper subgraph. More precisely, we first prove that for all prime powers $q \geq 3$, there exists a system $\mathscr{T}_q$ of triangles in $H_q$ such that no four span a $K_4$ in $H_q$, but every two-coloring of $E(H_q)$ induces a monochromatic triangle in $\mathscr{T}_q$. We then show that a certain random alteration of $H_q$ which destroys all of its $K_4$'s will, for large $q$, maintain the Ramsey property with high probability.

We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $α$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $α$ is almost-isometric to $M$.

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly $2d$ sets are needed.

This work presents the design and performance characterization of the optical calibration systems produced for the Pacific Ocean Neutrino Experiment (P-ONE), which target gain, energy and time calibration in the detector. These systems include novel light-pulse driver circuitry based on gallium nitride field-effect transistor technology and its application to directional and isotropic, self-monitoring optical calibration instruments. A total of 330 directional light pulsers and two isotropic, 17-inch calibration modules (P-CALs) were produced for the first P-ONE line. We present the designs and performance of both the directional and isotropic calibration devices and perform detailed optical characterizations of both full-production batches. In a wavelength range of $365 - 520\,$nm, our developed driver circuits achieve emission intensities up to $10^{11}\,$photons and pulse widths as small as $1.4\,$ns, respectively. Light-pulse drivers and self-monitoring electronics in the P-CAL were characterized using the same experimental setup, and the instrument's optical-isotropy design was optimized in combination with a dedicated GEANT4-based simulation framework. The optimized P-CAL achieves a simulated isotropy grade of $1.00 \pm 0.01$ across the entire $4π\,$solid angle range. These simulation investigations were explicitly confirmed by dedicated measurements in both air and water using two independent experimental setups, and we report the results. With this, a detailed performance estimate for deployed P-CAL modules in P-ONE was possible.

Let $E/\mathbb{Q}$ be an elliptic curve and let $ρ_E \colon \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{GL}(2, \widehat{\mathbb{Z}})$ be the adelic Galois representation attached to $E$. Much work has been done in recent years to study the image of $ρ_E$ (up to conjugation) as part of Mazur's so called ``Program B.'' In this paper, we describe and implement an efficient algorithm to compute the image of $ρ_E$ in $\operatorname{GL}(2, \widehat{\mathbb{Z}})$ (up to conjugation) for an elliptic curve $E/\mathbb{Q}$ with complex multiplication (CM) and $j$-invariant not $0$ or $1728$.

This letter presents a novel approach for \mbox{efficiently} computing time-index powered weighted sums of the form $\sum_{n=0}^{N-1} n^{K} v[n]$ using cascaded accumulators. Traditional direct computation requires $K{\times}N$ general multiplications, which become prohibitive for large $N$, while alternative strategies based on lookup tables or signal reversal require storing entire data blocks. By exploiting accumulator properties, the proposed method eliminates the need for such storage and reduces the multiplicative cost to only $K{+}1$ constant multiplications, enabling efficient real-time implementation. The approach is particularly useful when such sums need to be efficiently computed in sample-by-sample processing systems.

Solid-state scintillators are widely used in particle and applied physics due to their versatility and resistance to diverse environments and operating conditions. This broad range of applications calls for thorough characterization of scintillating crystals. Among these materials, cerium-doped yttrium aluminum garnet (YAG:Ce) is a promising scintillator owing to its favorable timing characteristics, high light yield, good mechanical properties, and chemical stability. In this work, we report a comprehensive experimental characterization of a YAG:Ce crystal exposed to both $γ$ and $alpha$ radiation. We extract the scintillation decay time and light yield, and study their evolution from room temperature down to approximately $-50 ^\circ$ C. We perform a detailed investigation of the quenching factor for \al particles in the energy range from about $6$ MeV down to $1$ MeV, finding a value that decreases from approximately $0.17$ to $0.10$. We also explore the possibility of pulse-shape discrimination based on the different signal evolution depending on the interaction type, demonstrating strong classification capabilities. These results provide a detailed assessment of the performance of \YAG for radiation-detection applications and offer insight into its potential use in environments requiring reliable particle identification and stable response across a wide range of operating conditions.

Single-photon sensitive detectors like Silicon Photomultipliers are widely used in many medical imaging applications. By using detectors with position resolutions, it is possible to build compact photodetector readouts with reduced number of channels, but still preserving position resolution and gamma-rays imaging capabilities. In this work, we present the advantage of using a Deep Neural Networks (DNNs) light position reconstruction applied to a 2x2 array of linearly-graded SiPMs (LG-SiPMs), to minimize the distortions on the reconstructed event maps. Our approach significantly enhances both the resolution and linearity of position detection compared to the nominal reconstruction formula based on the device architecture. Remarkably, the DNN-based reconstruction boosts the number of resolved areas (pixels) by a factor of 5.7 to 12.1 (depending the training splitting used) allowing for a higher level of precision and performance in light detection.

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representations of persistence diagrams are typically constructed extrinsically through embeddings into auxiliary spaces. For filtrations with finite indexing sets, the associated virtual persistence diagram group obtained by Grothendieck completion of the persistence diagram monoid is a finitely generated lattice. We define a phase map sending each persistence interval to a circular coordinate and a character map aggregating the phases of intervals in a virtual persistence diagram. We introduce heat damping on characters of virtual persistence diagram groups to suppress the unstable frequencies. We derive Lipschitz bounds for the resulting kernels and apply them in a synthetic segmentation experiment.

We show that the canonical actions of the Thompson group V and its generalizations on the Cantor set are not strongly ergodic. This implies that the associated crossed product von Neumann algebras are not full. This also yields a non-embedding result for the Thompson groups.

In this paper, we consider a contextual stochastic optimization problem in which unknown parameters follow distributions that depend on contextual covariates and decisions. The problem is motivated by transportation infrastructure decisions such as facility location or network design. In such high-stakes settings, decisions must often be communicated, justified, and reconsidered under alternative stakeholder requirements. To this end, we propose a framework for computing relative counterfactual explanations. These explanations identify the smallest changes in the covariates required for a solution to satisfy prescribed constraints while limiting the performance loss to a controlled level. Whereas relative explanations have been introduced in prior literature, to the best of our knowledge, this is the first work focusing on problems with binary decision variables and endogenous uncertainty. We propose a methodology that uses the Wasserstein distance as a regularization term in the objective. Beyond improving tractability, this regularization yields explanations with desirable structural properties: it produces sparser counterfactuals, induces smoother transitions in the underlying choice distributions, and keeps the counterfactual behavior close to realistic demand patterns. We illustrate the method using a choice-based competitive facility location problem and present numerical experiments that demonstrate its ability to efficiently compute sparse, plausible, and interpretable explanations. We further validate the framework on a real-world case study of electric vehicle charging station planning in Montreal, where the explanations reveal the minimal capacity investments and environmental conditions required to justify including a candidate location in the charging network.

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

A family of compact n-manifolds is locally combinatorially defined (LCD) if it can be specified by a finite number of local triangulations. We show that LCD is equivalent to the existence of a compact branched n-manifold W, such that the family is precisely those manifolds that immerse into W. In subsequent papers, the equivalence will be used to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD.

A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is satisfied, it is known that the associated graded Lie algebra of the arrangement group $G$ decomposes (in degrees greater than 1) as a direct product of free Lie algebras. It follows that the $I$-adic completion of the Alexander invariant $B(G)$ also decomposes as a direct sum of "local" invariants and the Chen ranks of $G$ are the sums of the local contributions. Moreover, if $B(G)$ is separated, then the degree 1 cohomology jump loci of the complement of $\mathcal{A}$ have only local components, and the algebraic monodromy of the Milnor fibration is trivial in degree 1.

Graph symmetries identify structural regularities and reduce the computational complexity of network analysis. In weighted graphs, however, exact automorphisms are rare because real-valued weights seldom coincide. We introduce a general framework for detecting approximate symmetries by aggregating weights into discrete categories, generating a sequence of coarser graphs on which classical automorphism analysis applies. The approximation path is fully configurable, based on interaction magnitudes, and can be matched to the empirical weight distribution. Applied to 250 empirical food webs using logarithmic aggregation, the method reveals that automorphisms emerge even at low approximation levels and almost always form small orbits. Orbit sizes rarely exceed two or three vertices, reflecting the combinatorial fragility of larger symmetric sets. Even so, symmetric vertices occupy diverse structural positions in the network and high connectivity does not imply asymmetry. The observation of just local permutations confirms the conclusions of trophic species and niche analysis. A case study demonstrates that automorphisms can also recover latent ecological structure. The minimal aggregation level at which two vertices become substitutable provides a quantitative measure of role similarity. The framework offers a principled, automorphism-based approach for quantifying similarity and redundancy in weighted complex networks.

Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomial ideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give a characterization for the Noetherian property of the Rees algebra of the family; and (b) present a combinatorial interpretation for the analytic spread of the family. We also apply these results to investigate the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants. We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals. In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

We extend discrete Morse-Bott theory to the setting of loop-free (or acyclic) categories. First of all, we state a homological version of Quillen's Theorem A in this context and introduce the notion of cellular categories. Second, we present a notion of vector field for loop-free categories. Third, we prove a homological collapsing theorem in the absence of critical objects in order to obtain the Morse inequalities. Examples are provided through the exposition. This answers partially a question by T. John: whether there is a Morse theory for loop-free (or acyclic) categories? [14].