AI for Science

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.

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 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.

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.

Elementary fuzzy Cellular Automata (CA) are known as continuous counterpart of elementary CA, which are 2-state CA, via the polynomial representation of local rules. In this paper, we first develop a new fuzzification methodology for $q$-state CA. It is based on the vector representation of $q$-state CA, that is, the $q$-states are assigned to the standard basis vectors of the $q$-dimensional real space and the local rule can be expressed by a tuple of $q$ polynomials. Then, the $q$-state vector-valued fuzzy CA are defined by expanding the set of the states to the convex hull of the standard basis vectors in the $q$-dimensional real space. The vector representation of states enables us to enumerate the number-conserving rules of 3-state vector-valued fuzzy CA in a systematic way.

Epicormic branches arise from dormant buds patterned during the growth of previous years. Dormant epicormic buds remain on the surface of trees, pushed outward from the pith during secondary growth, but maintaining vascular connections. Epicormic buds can be reactivated, either through natural processes or intentionally, to rejuvenate orchards and control tree architecture. Because epicormic structures are embedded within secondary growth, tomographic approaches are a useful method to study them and understand their development. We apply techniques from image processing to determine the locations of epicormic vascular traces embedded within secondary growth of sweet cherry (Prunus avium L.), revealing the juvenile phyllotactic pattern in the trunk of an adult tree. Techniques include breadth-first search to find the pith of the tree, edge detection to approximate the radius, and a conversion to polar coordinates to threshold and segment phyllotactic features. Intensity values from Magnetic Resonance Imaging (MRI) of the trunk are projected onto the surface of a perfect cylinder to find the locations of traces in the "boundary image". Mathematical phyllotaxy provides a means to capture the patterns in the boundary image by modeling phyllotactic parameters. Our cherry tree specimen has the conspicuous parastichy pair $(2,3)$, phyllotactic fraction 2/5, and divergence angle of approximately 143 degrees. The methods described not only provide a framework to study phyllotaxy, but for image processing of volumetric image data in plants. Our results have practical implications for orchard rejuvenation and directed approaches to influence tree architecture. The study of epicormic structures, which are hidden within secondary growth, using tomographic methods also opens the possibility of studying the genetic and environmental basis of such structures.

There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the $\ell^\infty$-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with $n$ labeled leaves as a point in $\mathbb{R}^{n(n+1)/2}$ known as the cophenetic vector, then comparing the two resulting Euclidean points using the $\ell^\infty$ distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the $\ell^\infty$-cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure; namely labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the $\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.

We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. To demonstrate the efficacy of our approach, we test our procedure on synthetic vehicular data generated by the Simulation of Urban Mobility package.

We estimate that data centers caused average retail electricity rates to fall modestly in the United States from 2015 to 2024 using an instrumental variables approach. Despite prevailing sentiment, the finding is consistent with economic reasoning: existing large power system fixed costs, economies of scale in transmission and distribution, and declining unit costs for generation imply that durable demand growth lowers average prices. We find patterns of economies of scale for transmission, distribution, and generation costs as well as within and across retail customer classes. We caution that future supply constraints could reverse the effect.

Diode-pumped solid-state (DPSS) lasers play a central role in modern photonics owing to their exceptional efficiency and ability to extend spectral coverage beyond the reach of semiconductor diodes. These attributes have enabled breakthroughs in precision metrology, quantum optics, and coherent communications. However, bringing the proven advantages of DPSS gain media such as Nd:YAG onto an integrated photonic platform has remained difficult, largely due to inefficient pump utilization and limited power-scaling in chip-scale implementations. Here, we demonstrate the first photonic-integrated Nd:YAG laser-amplifier system that overcomes these challenges with a micro-chip based master-oscillator-power-amplifier (μ-MOPA) architecture. The seed laser, employing a double-resonant microring resonator, could reach a threshold as low as 2.9 μW. The single-pass waveguide amplifier, when optimized separately, provides up to 46.6 dB small-signal gain. Combining the low-threshold seed with cascaded waveguide amplifiers, the integrated μ-MOPA delivers more than 12 dBm of amplified continuous-wave output power. These results establish Nd:YAG waveguide integration as a practical route to compact and high-performance solid-state light sources.

The expansion of the FIFA World Cup to 48 teams in 2026 introduces structural challenges in tournament design. To populate a 32-team knockout bracket from 12 groups of four, the current FIFA rules select the eight best third-placed teams using a global ranking across all groups. This global coupling creates several major problems: a combinatorial explosion of 495 possible bracket configurations; a fundamentally biased and unequal selection of third-placed qualifiers; lack of a clear path for group winners; vulnerability to collusion and ranking manipulation; and no guarantee of same-group separation beyond the first knockout round. We propose a simple unified solution called the four-section bracket (FSB) rule: split the 12 groups into four sections of three groups. All group winners, runners-up, and the two best third-placed teams in each section advance. Group winners remain in their home sections as local anchors, while lower-ranked qualifiers are transferred to other sections according to a fixed, symmetric rule. This structure guarantees same-group separation until the semifinal, protects the top eight group winners with a predictable knockout path, and reduces bracket complexity from 495 configurations to just one invariant topology per section, recovering the symmetry of the traditional 32-team format. We show substantial improvements in competitive fairness and scheduling predictability.

Coherent vacuum-ultraviolet (VUV) lasers are indispensable for precision measurement, quantum optics, and materials science. Recent high-resolution spectroscopy of the Th-229 nuclear clock transition near 148 nm highlights the urgent demand for intense, narrow-linewidth VUV lasers for advancing metrology and testing fundamental physics. However, existing VUV generation schemes typically require enhancement cavities [C. Zhang et al., Opt. Lett. 47, 5591-5594 (2022)], atomic resonances [Q. Xiao et al., Nature 650, 852-856 (2026)], or random quasi-phase-matched nonlinear crystals [V. Lal et al., Optica 12, 1971-1974 (2025)]. Here, we demonstrate a VUV frequency comb via cascaded frequency doubling of a 2400 nm Cr:ZnS comb to its 16th harmonic in nonlinear crystals. The final stage employs a bulk-grown, spatially uniform quasi-phase matched (QPM) crystal developed by IPG, combining VUV transparency, high $χ^2$ nonlinearity, and power scalability. Using this QPM crystal we generate a VUV frequency comb with 40 $μ$W average power (1 nW per mode at 80 MHz mode spacing) with a conversion efficiency order of magnitude higher than other known methods. These results establish a scalable route to compact VUV sources via direct frequency doubling, opening a path toward a robust continuous-wave nuclear clock laser.

Modern microscopy systems integrate heterogeneous hardware devices that require dedicated software for coordination. However, high-performance C++ implementations of microscopy control software remain scarce. We present ScopeOne, a C++ and Qt-based microscopy control software built on the MicroManager hardware abstraction layer. Through process isolation and shared memory, ScopeOne achieves simultaneous multi-camera preview with real-time image processing, while maintaining full compatibility with the μManager device ecosystem.

The J-PARC E16 experiment aims to search for signatures of chiral symmetry restoration. It studies in-medium modifications of vector mesons that decay via the dielectron channel. The measurements use a high-intensity 30 GeV proton beam with C and Cu targets at rates up to 10 MHz. To achieve this, the experiment upgrades its tracking, by introducing innermost detector modules constructed with the same technology and procedures as the modules of the Silicon Tracking System (STS) of the Compressed Baryonic Matter (CBM) experiment at Facility for Antiproton and Ion Research (FAIR). A total of 15 modules were assembled, tested, characterized and then installed in the E16 detector setup. The detector was commissioned in a beam test experiment at Tsukuba, where the detector modules could be exposed to a 3 GeV electron beam. In preparation for the beam test the modules were characterized and calibrated, and performance studies were accomplished to assess the quality of the setup. During beamtime, three modules were operated and illuminated in two planes by the electron beam. This paper presents the results of the construction, characterization, commissioning, and operation of the E16-STS modules in beam test experiments.

We show that the non-Hermitian skin effect (NHSE) can enhance pairing correlations in moiré Hubbard systems through a channel-selective mechanism: skin-induced localization amplifies the boundary density of states, strengthening local pairing tendencies within an intermediate ``golden window'' of non-reciprocity $γ\in[0.5,1.2]\,t$. Using exact diagonalization of the non-Hermitian Hubbard model on triangular lattices with open boundaries, we map the $(U,γ)$ phase diagram. The double occupancy $D(γ)$ exhibits non-monotonic behavior -- rising by up to 21\% then declining -- reflecting a competition between NHSE-enhanced boundary pairing and over-localization. A decomposition of the pairing susceptibility $χ_{\mathrm{SC}}$ on the $3\times3$ cluster reveals that the NHSE acts \emph{channel-selectively}: it enhances on-site pairing ($+21\%$) while simultaneously suppressing competing antiferromagnetic correlations (22\% reduction), so that the total pairing susceptibility, dominated by the on-site channel, grows by $+98\%$ on that cluster. These trends are corroborated by an independent non-Hermitian DMRG calculation and establish an enhancement of finite-cluster pairing correlations rather than trivial density redistribution. We do not claim long-range superconducting order. A BCS scaling estimate converts the same pairing-response signal into a dome-shaped $T_c(γ)$ fingerprint, suggesting an experimentally distinguishable response in coherent-drive versus reservoir-dominated moiré devices.

We revisit the charge-1/3 anyon fluid obtained by doping the $ν= 2/3$ Jain fractional Chern insulator (FCI). In the standard composite fermion description, the doped anyons fractionalize into three translation-related flavors of secondary composite fermions, whose gauge-mediated interactions drive a robust inter-flavor pairing instability. In our previous work, we analyzed a flavor-asymmetric paired state and obtained a charge-ordered Fermi liquid. Inspired by a recent paper, we consider an alternative flavor-symmetric paired state and show that it is an SC* state: a charge-$2e$ condensate that coexists with residual $\mathbb{Z}_2$ topological order. The weak and strong pairing regimes share the same intrinsic topological order but differ in chiral central charge, giving $c_- = 7/2$ and $c_- = 2$. We further show how other proposed effective field theories fit within the same composite fermion description, and argue that across the doping driven FCI-to-superconductor transition, localized anyons evolve into Bogoliubov quasiparticles rather than vortices. At low doping, we identify an approximate SU(3)-symmetric regime in which the system instead realizes a non-Fermi liquid $\mathbb{Z}_3$ Orthogonal Metal with three charge-1/3 fermion pockets and no sharp electron quasiparticle. Finally, we comment on the energetics of various possible ground states and discuss implications for experiments in moire materials.

Rhombohedral multilayer graphene aligned with hexagonal boron nitride (RMG-hBN) hosts correlated Chern phases, but the microscopic role of hBN stacking remains unclear, especially when the active carriers are displaced away from the moiré interface. Using Hartree-Fock calculations over layer numbers, twist angles, displacement fields, fillings, and hBN alignments, we show that correlated insulators are most robust at small twist angles and intermediate layer number ($N\simeq 6$), where bandwidth suppression is balanced by layer delocalization of the wavefunctions of the active carriers. Under moiré-distant conditions at filling $ν=1$, the topology of the insulating state is strongly biased by charge imprinting: the hBN alignment shapes the occupied valence-band charge texture near the interface via moiré potential, which acts through long-range Coulomb interactions as a remote electrostatic template for doped conduction electrons. Depending on the alignment, this template favors either triangular charge localization associated with trivial insulators or honeycomb-like charge networks compatible with Chern insulators. Our results identify valence-band charge textures as a microscopic route by which a remote moiré interface controls correlated topology in multilayer graphene.

Magnetic quantum oscillations are usually periodic in inverse magnetic field, and their amplitude can show beating when two nearby frequencies interfere. In graphene-based hexagonal systems, such beating can arise from strain-induced pseudomagnetic fields, unequal valley populations, valley-dependent energy shifts, spin-orbit coupling-induced band splitting, or Kekulé distortions. Here, we show that the carrier density and magnetic field dependence of the beating nodes can distinguish these mechanisms. Starting from Onsager's quantization relation, we derive scaling relations for the critical carrier density $N_c$ for the beating nodes as a function of critical magnetic field $B_c$. A pseudomagnetic field gives $N_c\propto B_c^2$, whereas a density-independent valley imbalance gives $N_c\propto B_c$. A constant Dirac-band energy splitting by Zeeman-like spin-orbit coupling also gives quadratic field scaling, but with a different node sequence: $N_{c,j}\propto(2j+1)B_{c,j}^2$ for a pseudomagnetic field and $N_{c,j}\propto(2j+1)^2B_{c,j}^2$ for energy splitting, where $j$ labels the beating node indices. These results provide quantitative constraints on different microscopic origins of valley- and spin-dependent band splittings in graphene-based systems.

Active particles exhibit self-propulsion, leading to transport behavior that differs fundamentally from passive Brownian motion. In confined or structured domains, activity strongly influence escape probabilities and first-passage behavior. Understanding these effects is essential for describing transport in biological microenvironments, microfluidic devices, and heterogeneous media. In this work, leveraging the backward Fokker--Planck equation, we investigate the splitting probability of chiral active Brownian particles in confined domains, focusing on both a one-dimensional interval and a two-dimensional corrugated channel. Analytical solutions are derived for the one-dimensional case in various asymptotic regimes. In corrugated channels with small aspect ratios, we develop a Fick--Jacobs reduction that yields effective transport equations along the axial direction, whereas for finite aspect ratios, the splitting dynamics are characterized numerically. We demonstrate how channel geometry, particle activity, and chirality modulate the likelihood of escape through different boundaries. Our results provide quantitative predictions for the transport of active matter in complex environments and highlight the interplay between confinement and activity.

While proper orthogonal decomposition (POD)-based surrogates are widely explored for hydrodynamic applications, the use of Koopman autoencoders for real-world coastal-ocean modelling remains relatively limited. This paper introduces a flexible Koopman autoencoder formulation that incorporates meteorological forcings and boundary conditions, and systematically compares its performance against POD-based surrogates. The Koopman autoencoder employs a learned linear temporal operator in latent space, enabling eigenvalue regularization to promote temporal stability. This strategy is evaluated alongside temporal unrolling techniques for achieving stable and accurate long-term predictions. The models are assessed on three test cases spanning distinct dynamical regimes, with prediction horizons up to one year at 30-minute temporal resolution. Across all cases, the reduced order surrogates with temporal unrolling achieve high accuracy with relative root-mean-squared-errors of 0.0068-0.14 and $R^2$-values of 0.61-0.995, where prediction errors are largest for current velocities, and smallest for water surface elevations. In two of the three cases, the Koopman Autoencoder have higher accuracy than the POD-based surrogates. Comparing to in-situ observations, the surrogate yields -0.64% to 12% increase in water surface elevation prediction error when compared to prediction errors of the physics-based model. These error levels, corresponding to a few centimeters, are acceptable for many practical applications, while inference speed-ups of 300-1400x enables workflows such as ensemble forecasting and long climate simulations for coastal-ocean modelling.

These notes provide a detailed and self-contained exposition of basic theory of CP maps and continuous-time Markovian evolution.The infinite-dimensional (separable) setting is handled as an extension of the finite-dimensional one.The treatment stands on two legs.For the finite-dimensional part, a basis-free version of the Choi-Jamiolkowski isomorphism called simply Jamiolkowski transform.And, for the extension, the ground matrix element topology (GMET), which does for the superoperators on trace-class operators what the weak-operator topology does for bounded operators on a Hilbert space. Background in open quantum systems or quantum information theory is not assumed.

Max-plus algebra is a semiring with addition $a\oplus b = \max(a,b)$ and multiplication $a\otimes b = a+b$. It is applied in cases, such as combinatorial optimization and discrete event systems. We consider the power of max-plus square matrices, which is equivalent to obtaining the all-pair maximum weight paths with a fixed length in the corresponding weighted digraph. Each $n$-by-$n$ matrix admits the CSR expansion that decomposes the matrix into a sum of at most $n$ periodic terms after $O(n^{2})$ times of powers. In this study, we propose an $O(n(m+n \log n))$ time algorithm for the CSR expansion, where $m$ is the number of nonzero entries in the matrix, which improves the $O(n^{4} \log n)$ algorithm known for this problem. Our algorithm is based on finding the roots of the characteristic polynomial of the max-plus matrix. These roots play a similar role to the eigenvalues of the matrix, and become the growth rates of the terms in the CSR expansion.

In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.

We present a novel approach to implementing all-optical Rectified Linear Unit (ReLU) activation functions using compact doubly-resonant cavities with dimensions of approximately $10\,μ\mathrm{m}$. Our design leverages $χ^{(2)}$ nonlinear processes within carefully engineered photonic structures that simultaneously resonate at both fundamental and second-harmonic frequencies. By exploiting the phase-sensitive nature of second-harmonic generation, we demonstrate an optical analog to the ReLU function, achieving femtojoule-level activation energy-comparable to state-of-the-art approaches-while reducing device footprint by two orders of magnitude compared to previous implementations. We develop the theoretical framework using coupled-mode theory and validate it through rigorous finite-difference time-domain simulations. Beyond ReLU, we show that the same physical structure can implement alternative activation functions such as ELU and GELU through simple adjustments to input conditions. Neural network simulations demonstrate that our proposed optical activation functions achieve classification accuracy within $0.4\%$ of ideal electronic implementations while offering significant advantages in energy efficiency and processing speed. This work represents a significant advancement toward realizing energy-efficient, high-density optical neural networks for next-generation artificial intelligence hardware.

Soft and active condensed matter represent a class of fascinating materials that we encounter in our everyday lives -- and constitute life itself. Control signals interact with the dynamics of these systems, and this influence is formalized in control theory and optimal control. Recent advances have employed various control-theoretical methods to design desired dynamics, properties, and functionality. Here we provide an introduction to optimal control aimed at physicists working with soft and active matter. We describe two main categories of control, feedforward control and feedback control, and their corresponding optimal control methods. We emphasize their parallels to Lagrangian and Hamiltonian mechanics, and provide a worked example problem. Finally, we review recent studies of control in soft, active, and related systems. Applying control theory to soft, active, and living systems will lead to an improved understanding of the signal processing, information flows, and actuation that underlie the physics of life.

In this paper, we prove a Heintze-Karcher type inequality for capillary hypersurfaces supported on various hypersurfaces in the hyperbolic space. The equality case only occurs on capillary totally umbilical hypersurfaces. Then we apply this result to prove the Alexandrov type theorem for embedded capillary hypersurfaces in the hyperbolic space. In addition, we prove some other rigidity results for capillary hypersurfaces supported on totally geodesic plane in $\mathbb B^{n+1}_+$.

We investigate the realization of a time-reversal-broken Weyl semimetal in Floquet synthetic dimensions generated by two incommensurate drives, in the spirit of topological frequency conversion in driven synthetic lattices PRX 7, 041008 (2017). The system is described by a one-dimensional lattice model in a mixed $(1~\mathrm{real}+2~\mathrm{synthetic})$-dimensional setting, where the driving phases act as synthetic momenta and generate Weyl points in the mixed Floquet band structure. Using the topology associated with these band degeneracies, we analyze the energy transfer between the two drives. We find that the mixed Floquet lattice captures the Weyl-semimetal topology only in a momentum-resolved sense: for fixed real momentum $k_x$, the power transfer measures the $k_x$-resolved Chern number and detects the separation of the Weyl nodes. However, the full real-space response is qualitatively different. The total power transfer does not reproduce the static Weyl-semimetal phase diagram, but instead follows an effective Rice-Mele-type pumping structure. Thus, in contrast to fully gapped topological insulators, gapless semimetallic phases do not straightforwardly translate to Floquet synthetic dimensions. Our results reveal a distinct dynamical phase structure of driven Weyl systems and establish mixed Floquet lattices as a platform for exploring non-equilibrium gapless topology.