Holographic multiple-input multiple-output (MIMO) has emerged as a key enabler for 6G networks, yet conventional planar implementations suffer from spatial correlation and mutual coupling at sub-wavelength spacing, which fundamentally limit the effective degrees of freedom (EDOF) and channel capacity. Three-dimensional (3-D) holographic MIMO offers a pathway to overcome these constraints by exploiting volumetric array configurations that enlarge the effective aperture and unlock additional spatial modes. This work presents the first systematic evaluation that jointly incorporates electromagnetic (EM) characteristics, such as mutual coupling and radiation efficiency, into the analysis of 3-D arrays under Clarke, Kronecker, and standardized 3rd Generation Partnership Project (3GPP) channel models. Analytical derivations and full-wave simulations demonstrate that 3-D architectures achieve higher EDOF, narrower beamwidths, and notable capacity improvements compared with planar baselines. In 3GPP urban macro channels with horizontal element spacing of 0.3 lambda, 3-D configurations yield approximately 20% capacity improvement over conventional 2-D arrays, confirming the robustness and scalability of volumetric designs under realistic conditions. These findings bridge the gap between theoretical feasibility and practical deployment, offering design guidance for next-generation 6G base station arrays.

Herein, we conduct a comprehensive investigation of Hexa-graphyne (HXGY), a planar carbon allotrope formed by distorted hexagonal and rectangular rings incorporating sp and sp$^2$-hybridized carbon atoms. First-principles calculations confirm its energetic, dynamical and thermal stability (up to at least 1000 K). Regarding its band structure, this material exhibits a semimetallic nature. It exhibits high mechanical compliance, with a Young's modulus approximately 13 times lower and a Poisson's ratio nearly 4 times higher than those of graphene. The optical response is marked by strong ultraviolet absorption, high infrared reflectivity, and pronounced transparency in the visible-light range. Raman and infrared spectra exhibit sharp and well-separated peaks, providing a clear signature of acetylenic linkage stretching vibrations. Nanoribbon structures derived from HXGY show distinct electronic behaviors depending on the edge termination type and width. These findings highlight the HXGY potential for nanoelectronic and optoelectronic applications.

We study the perturbative path integral of Chern-Simons theory (the effective BV action on zero-modes) in Lorenz gauge, expanded around a (possibly non-acyclic) flat connection, as a family over the smooth irreducible stratum $\mathcal{M}' \subset \mathcal{M}$ of the moduli space of flat connections. We prove that it is horizontal with respect to the Grothendieck connection up to a BV-exact term. From it, we construct a volume form on $\mathcal{M'}$ - the "global partition function" - whose cohomology class is independent of the metric, and so is a 3-manifold invariant. As an element of the construction, we construct an extension of the perturbative partition function to a nonhomogeneous form on the space of triples $(A,A',g)$ consisting of (1) a "kinetic" flat connection $A$ around which Chern-Simons action is expanded, (2) a "gauge-fixing" flat connection $A'$, (3) a metric $g$. This extension is horizontal with respect to an appropriate Gauss-Manin superconnection (which involves the BV operator as a degree zero component).

We present the first lattice QCD determination of the electromagnetic form factors of the exotic tetraquark $T_{bb} \ (bb \bar u \bar d)$ with quantum numbers $I( J^P ) = 0( 1^+ )$. The extracted form factors encode information about its internal structure, including the charge distribution and the magnetic dipole moments, determined separately for the light and heavy quarks. Our results provide evidence in favor of it being a bound state consisting of a compact heavy diquark $[bb]$ in a color-antitriplet with spin one, and a light antidiquark $[\bar u \bar d]$ in a color-triplet with spin zero. The charge radius of $T_{bb}$ is found to be significantly smaller than the combined charge radii of $B$ and $B^*$ mesons. These two comprise the lowest-lying threshold $BB^*$ in the channel we are considering, and their electric charge form factors are also determined. The computations were performed on a single CLS ensemble with $N_f = 2+1$ dynamical quarks and a lattice spacing of approximately $a \approx0.064 \ \mathrm{fm}$ at the pion mass $m_π\approx 290 \ \mathrm{MeV}$.

Nuclear-spin conversion in molecular hydrogen is governed by strict symmetry rules that typically require magnetic fields or catalytic surfaces to break. Here we demonstrate that the intrinsic tensor composition of a non-magnetic molecular crystal field can impose and relax these rules without external fields. High-resolution infrared spectra of H$_2$ in crystalline CO$_2$ reveal large rank-2 (quadrupolar) crystal-field splittings of the $m$ sublevels, while nuclear-spin conversion occurs only through $Δm = 0$ channels. Replacing CO$_2$ with polar N$_2$O introduces rank-1 (dipole) components that partially open $Δm \neq 0$ pathways, while incorporation of paramagnetic NO$_2$ fully lifts the restriction. These results establish a direct correspondence between crystal-field tensor rank and nuclear-spin dynamics, introducing a general symmetry-based framework for designing and controlling spin-isomer populations and quantum-state connectivity in molecular solids.

In crystalline systems with a superstructure, the electron dispersion can form a nontrivial covering of the Brillouin zone. It is proved that the number of sheets in this covering and its monodromy are topological invariants under ambient isotopy. As a concrete manifestation of this nontrivial topology, we analyze three-sublattice models for 120$^\circ$-ordered helimagnets in one, two, and three dimensions. The two-dimensional system exhibits unconventional $f$-wave magnetism and a specific topological metal state characterized by a spin-textured, one-sheeted Fermi surface. The observable transport signatures of the topological metal and its potential experimental realization are briefly discussed.

We investigate scalar-induced stochastic gravitational waves from adiabatic curvature perturbations sourced by a spectator field via the modulated reheating mechanism. We consider a spectator scalar with Higgs-like couplings and inflaton decay via shift symmetric dimension-five operators. The spectator is assumed to be in the Sitter vacuum and it sources blue-tilted, strongly non-Gaussian curvature perturbations which can dominate the spectrum on small scales $k \gg \rm{Mpc}^{-1}$. We find that the setup could generate a gravitational wave signal testable by surveys like BBO and DECIGO but only for large coupling values not expected in low-energy particle physics setups that can be perturbatively extrapolated up to the inflationary scale.

Originally envisioned as a solution for the capacity crunch in telecommunications networks, multicore fibers (MCF) are contributing to scientific fields beyond telecom, such as sensing and metrology. Confined within the same cladding, the cores of MCF have a high degree of noise correlation which can be harnessed for a variety of applications. Here, we investigate MCF as a solution to the challenging problem of quantum and classical light co-existence in quantum networks by operating the quantum and stabilization light in separate but highly correlated cores of a 7-core MCF. Over 40 km of spooled fiber, we achieved 100 attosecond integrated jitter on one core by using phase information derived from another core. This allows for 100% duty cycle on a quantum channel while maintaining a low spurious photon rate from crosstalk between stabilization and quantum channels. With cycle-slip-free stabilization over 6 hours, frequency detuning between designated stabilization and quantum channels, and an additional 40 dB rejection of noise photons provided by the low optical crosstalk between cores, we achieved a Raman scattering-induced spurious photon rate of only 0.01 photons/s in 100 GHz bandwidth. Our results with MCF are a promising approach to ultra-stable quantum networks with 100% duty cycle on the quantum channel.

We present a systematic study of bound relations between different electronic properties of magnetic crystals: electron density, effective mass, orbital magnetization, localization length, Chern invariant, and electric susceptibility. All relations are satisfied for a group of low-lying bands, while some remain valid for upper bands. New results include a lower bound on the electric susceptibility of Chern insulators, and an upper bound on the sum-rule part of the orbital magnetization. In addition, bounds involving the Chern invariant are generalized from two dimensions (Chern number) to three (Chern vector). Bound relations are established for metals as well as insulators, and are illustrated for model systems. The manner in which they approach saturation in a model Chern insulator with tunable flat bands is analyzed in terms of the optical absorption spectrum.

This paper presents an adaptive air transit network leveraging modular aerial pods and artificial intelligence (AI) to address urban mobility challenges. Passenger demand, forecasted from AI models, serves as input parameters for a Mixed-Integer Nonlinear Programming (MINLP) optimization model that dynamically adjusts pod dispatch schedules and train lengths in response to demand variations. The results reveal a complex interplay of factors, including demand levels, headway bounds, train configurations, and fleet sizes, which collectively influence network performance and service quality. The proposed system demonstrates the importance of dynamic adjustments, where modularity mitigates capacity bottlenecks and improves operational efficiency. Additionally, the framework enhances energy efficiency and optimizes resource utilization through flexible and adaptive scheduling. This framework provides a foundation for a responsive and sustainable urban air mobility solution, supporting the shift from static planning to agile, data-driven operations.

This paper aims to introduce a mathematical model to solve the time-dependent electric vehicles routing problem in shared travels. Shared mobility has gained significance recently due to its contribution to the alleviation of traffic congestion and air pollution. In this study, a MIP model has been developed and solved for a toy problem using the CPLEX solver to indicate the efficiency of shared mobility compared to private mode. We have considered practical constraints, such as considering the traffic congestion throughout the day by assigning a time-dependent step function to the network's links altering travel times in the peak and off-peak hours, nonlinear charging function, vehicles' queue at charging stations, partial charging possibility, different charging infrastructures, and passengers' desired timewindows, to make the results of the model more applicable in the real world. Among the important results of this presented model according to the solved example, we can mention the reduction of mileage per vehicle in personal mode from 56.42 km to 46.83 km in shared mode while the total travel time per vehicle has only increased slightly from 84.63 min/veh to 89.32 min/veh, respectively. This suggests that the typical problems related to private travel can be offset by shared travel without losing the comfort and convenience of the former.

Input-output theory is a well-known tool in quantum optics and ubiquitous in the description of quantum systems probed by light. Owing to the generality of the setup it describes, the theory finds application in a wide variety of experiments in circuit and cavity QED. We present an approach to input-output theory using the Schwinger-Keldysh path integral formalism that gives us direct access to the full output field statistics such as the first and second order coherence functions. By making the rich toolbox of non-equilibrium quantum field theory accessible, our formalism greatly simplifies the treatment of nonlinear systems and provides a uniform way of obtaining perturbative results. We showcase this particular strength by computing the output field statistics of a Kerr nonlinear oscillator at finite temperatures through the use of diagrams and diagram summation techniques. We find a reduction in reflection that is not due to photon leakage but rather associated to the squeezing of the output light.

A novel random field model or the reconstruction of turbulent velocity fluctuations from inhomogeneous characteristic flow quantities in terms of stochastic Fourier-type integrals has recently been introduced and analyzed by the authors. This article concerns the numerical discretization and implementation of the model and discusses its key features by means of numerical simulations. We present a suitable discretization scheme that combines a randomized quadrature method for stochastic integrals with a local linearization of the non-uniform advection of the turbulent structures by the mean flow. The convergence of the scheme towards the continuous model is verified analytically. Moreover, we describe an efficient algorithmic implementation that allows for flexible local evaluations of the simulated turbulence field. The main features of the model are illustrated by a variety of simulation results, each highlighting specific aspects such as the influence of the inhomogeneous model parameters on the generated fluctuations, spatio-temporal ergodicity properties under inhomogeneous flow conditions, and the validity of Kolmogorov's two-thirds law in dependence on the local turbulence Reynolds number.

We derive an explicit formula for the holonomy $R$-matrix of quantum $\mathfrak{sl}_2$ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

We present an exact, purely geometric account of the interior dynamics of Schwarzschild black holes, formulated without invoking any specific gravitational theory and free of additional charges beyond the total mass ${\cal M}$. We show that dynamical evolution generically produces new singularities, absent in the static case, whose resolution imposes highly restrictive conditions on gravitational collapse.

Active galactic nuclei (AGN) jets are powerful drivers of galaxy evolution, depositing energy and momentum into the circumgalactic and intracluster medium (CGM/ICM) and regulating gas cooling and star formation. We investigate the dynamics of jet evolution in the self-similar regime using the smoothed particle hydrodynamics (SPH) code GADGET4-Osaka, systematically vary jet-launching schemes, artificial-viscosity prescriptions, mass resolution, and jet lifetimes and compare the results with grid-based simulation. Our analysis combines quantitative diagnostics of jet size and energetics with detailed morphological and thermodynamic characterizations from slice maps and phase diagrams. We find that jet lobe growth follows analytic self-similar scaling relations and converges with resolution, but is highly sensitive to the choice of artificial viscosity. While the overall jet size tracks self-similar predictions, the partitioning of thermal and kinetic energy departs significantly from the idealized picture, reflecting enhanced dissipation and mixing, which is consistent with the jet propagation in grid-based simulations. These results establish robust benchmarks for SPH-based jet modeling, provide insight into the physical and numerical factors shaping jet--medium interactions, and lay the groundwork for future studies of AGN feedback in realistic galactic and cluster environments.

Graph Signal Processing (GSP) extends classical signal processing to signals defined on graphs, enabling filtering, spectral analysis, and sampling of data generated by networks of various kinds. Graphon Signal Processing (GnSP) develops this framework further by employing the theory of graphons. Graphons are measurable functions on the unit square that represent graphs and limits of convergent graph sequences. The use of graphons provides stability of GSP methods to stochastic variability in network data and improves computational efficiency for very large networks. We use GnSP to address the stimulus identification problem (SIP) in computational and biological neural networks. The SIP is an inverse problem that aims to infer the unknown stimulus s from the observed network output f. We first validate the approach in spiking neural network simulations and then analyze calcium imaging recordings. Graphon-based spectral projections yield trial-invariant, lowdimensional embeddings that improve stimulus classification over Principal Component Analysis and discrete GSP baselines. The embeddings remain stable under variations in network stochasticity, providing robustness to different network sizes and noise levels. To the best of our knowledge, this is the first application of GnSP to biological neural networks, opening new avenues for graphon-based analysis in neuroscience.

Through extensive self-consistent Hartree-Fock calculations in a tight-binding model of twisted bilayer graphene (TBG), we show that many-body effects lead to a considerable increase of the bandwidth of the flat bands and, concomitantly, to a shift of the magic angle (defined by the condition of minimum bandwidth). Specifically, we predict a shift from the $\textit{ab initio}$ magic angle of $0.99^\circ$ to a renormalized value of $0.88^\circ$ for a TBG sample suspended between metallic gates with a gate-to-gate distance of $10 \text{ nm}$. We derive analytical expressions for the renormalized Fermi velocity and interlayer couplings, finding good agreement with the numerical results, and investigate the convergence toward the numerical solutions with respect to the number of renormalized couplings of a generalized Bistritzer-MacDonald (BM) model. Using the derived analytical formulas, we demonstrate the possibility of tuning the flat bands via different dielectric environments and gate geometries in the experiments. Furthermore, we predict a significant enhancement of the flat-band Fermi velocity at intermediate twist angles relative to the bare value, and propose measurements in this range as a probe of the effective couplings of TBG. Our results imply a change of paradigm whereby the maximum $T_c$ for superconductivity would correspond to a condition of small but not minimum bandwidth.

We investigate the Nagaoka-Thouless (NT) ferromagnetic instability in the strongly interacting $t$-$t'$ Hubbard model by continuously breaking particle-hole symmetry on a tunable square-triangular lattice geometry. We use an analytic approach to show that the fully spin-polarized state becomes unstable to a metastable spin-polaron when the kinetic frustration $t'/t$ exceeds a critical, dimension-dependent value. Large-scale density matrix renormalization group simulations reveal a quantum phase transition from the NT ferromagnet to a spiral spin-density wave, which evolves continuously into the Haerter-Shastry antiferromagnet in the large-frustration limit. Remarkably, this transition remains robust at low but finite hole density, making it accessible in cold-atom and moiré Hubbard platforms under strong interactions. A variational analysis further captures the instability mechanism at finite density via frustration-induced magnon band deformation.

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $λ=(λ_1, \dots, λ_m)$ into $m$ parts, with $λ_1\ge \dots \ge λ_m$, such that either $λ_1=1$ or $λ_{k-1}>λ_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

We introduce OpenLifelogQA, a large-scale open-ended lifelog QA dataset constructed from 18 months of multimodal lifelog data. Lifelogging is the passive collection and analysis of personal daily activities using wearable devices, producing rich multimodal data such as images, locations, and biometrics. Question answering (QA) over lifelog data enables users to interactively query their own experiences, supporting applications in memory support, lifestyle analysis, and personal assistance. OpenLifelogQA contains 14,187 Q&A pairs spanning multiple question types and difficulty levels, designed to support robust evaluation in realistic settings. Compared with prior resources, OpenLifelogQA offers greater diversity and practicality for real-world applications. To establish baselines, we evaluate the LLaVA-NeXT-Interleave 7B model, achieving 89.7% BERTScore, 25.87% ROUGE-L, and an average LLM Score of 3.97. By releasing OpenLifelogQA, we aim to promote future research on lifelog technologies, paving the way for personal lifelog assistants capable of memory augmentation, healthcare support, and lifestyle coaching.

The study of biomolecular folding has been greatly advanced by single-molecule force spectroscopy (SMFS), which enables the observation of the dynamics of individual molecules. However, extracting quantitative models of fundamental properties such as folding landscapes from SMFS data is very challenging due to instrumental noise, linker artifacts, and the inherent stochasticity of the process, often requiring extensive datasets and complex calibration. Here, we introduce a framework based on simulation-based inference (SBI) that overcomes these limitations by integrating physics-based modeling with deep learning. We first apply this framework to analyze constant-force measurements of a DNA hairpin. From a single experimental trajectory of only two seconds, we successfully reconstruct the hairpin's free energy landscape and folding dynamics, obtaining results in close agreement with established deconvolution methods that require 10-100 times more data. Furthermore, we demonstrate the generality of our approach by applying it to a riboswitch aptamer featuring multiple states and tertiary contacts, resolving the profile of a landscape featuring four metastable states from a single trajectory. The Bayesian nature of this approach robustly quantifies uncertainties for all inferred parameters, including diffusion coefficients and linker stiffness, without needing independent measurements of instrument properties. The inferred models are predictive, generating simulated trajectories that quantitatively reproduce experimental thermodynamics and kinetics. The ability to derive statistically robust models from minimal datasets is crucial for investigating complex biomolecular systems where extensive data collection is impractical, paving the way for novel applications of SMFS.

A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $α\in H^2(M,\mathbb R)$ admit a symplectic representative. The collection of all such classes, the symplectic cone $\mathcal C_M$, is a basic smooth invariant of $M$. This paper describes the symplectic cone for elliptic surfaces with positive Euler number.

The discrepancy between the early-time estimation and late-time observation on the cosmic baryon content - the 'missing baryon problem' - is a longstanding problem in cosmology. Although recent studies with fast radio bursts (FRBs) have largely addressed this discrepancy, the precise spatial distribution of these baryons remains uncertain due to the effect of galaxy feedback. Cosmological hydrodynamical simulations such as Simba have shown that the partitioning of baryons between the intergalactic medium (IGM) and haloes is sensitive to feedback models, motivating the connection of baryon distribution with feedback physics. With the Simba simulation suite, this study investigates how feedback affects the distribution of matter within large-scale cosmic structures, with implications for FRB foreground modeling. We apply the T-web method to classify the cosmic web into different structures: knots, filaments, sheets, and voids. We then analyze how the different feedback variants of Simba affect the distribution of matter within each structure. Our results show that in Simba, the fractions of IGM gas in different cosmic web structures vary only a few percent under different feedback models. However, jet feedback produces noticeable changes in the gas distribution within structures, enhancing the diffuse IGM on the outskirts of filaments and knots. This research provides a new perspective on the impact of feedback on the IGM and motivates a refined data model for the FRB foreground mapping.

Symmetries are a key tool in understanding quantum systems, and, among many other things, can be exploited to increase the efficiency of numerical simulations of quantum dynamics. Disordered systems usually feature reduced symmetries and additionally require averaging over many realizations, making their numerical study computationally demanding. However, when studying quantities linear in the time-evolved state, i.e. expectation values of observables, one can apply the averaging procedure to the time evolution operator, resulting in an effective dynamical map, which restores symmetry at the level of superoperators. In this work, we develop schemes for efficiently constructing symmetric sectors of the disorder-averaged dynamical map using short-time and weak-disorder expansions. To benchmark the method, we apply it to an Ising model with random all-to-all interactions in the presence of a transverse field. After disorder averaging, this system becomes effectively permutation-invariant, and thus the size of the symmetric subspace scales polynomially in the number of spins allowing for the simulation of large systems.

We present an analysis of the lepton-number and lepton-flavour-violating process of muon-to-positron conversion $μ^- + N \rightarrow e^+ + N'$, in the presence of an ultralight scalar dark matter (ULSDM) field which couples to neutrinos. The ULSDM contributes to the effective off-diagonal Majorana mass $ m_{μe}$, therefore amplifying the rate of muon-to-positron conversion to experimentally observable levels. Using existing bounds from SINDRUM II, COMET, and Mu2e experiments, we derive novel constraints on the flavour-off-diagonal couplings of neutrinos to ULSDM. Our work reveals that upcoming experiments can provide stronger sensitivity to these new couplings than bounds arising from cosmological surveys and terrestrial experiments.

In this article, we analyze an "impossible measurement" scenario presented by Sorkin. This scenario involving a joint measurement on spacelike separated systems in an intermediary region has widely been discussed in the quantum field theory measurement literature. We analyze the non-relativistic version of this paradoxical measurement scenario in full detail and give explicit bounds for the amount of signaling present. We also discuss the conditions under which no extraneous signaling occurs.

The cuprate pseudogap phase displays Fermi arc spectral weight in photoemission and scanning tunneling microscopy (STM), while recent magnetotransport observations yield evidence for the existence of hole pockets of fractional area $p/8$, where $p$ is the doping density. We present a Monte Carlo study of a thermal SU(2) lattice gauge theory which can reconcile these observations. Our simulation includes the SU(2) gauge field $U$ of a $π$-flux spin liquid, and a SU(2) fundamental charge $e$ Higgs boson $B$. There is a Yukawa coupling between $B$, the fermionic spinons of the spin liquid, and the hole pockets of a fractionalized Fermi liquid. At the higher temperatures of the pseudogap, the finite-doping sign problem is evaded by including only thermal fluctuations of $B$ and $U$, while the fermions are diagonalized exactly for each boson background. Our study also yields a fractionalized description of intertwined orders at lower temperatures, including the onset of $d$-wave superconductivity by the expulsion of vortices with flux $h/(2e)$, each with charge-order halos. We discuss conditions under which quantum oscillations in the density of states from hole pockets of area $p/8$ could be observable in clean under-hole-doped cuprates.

Compact binaries with large mass asymmetries - such as Extreme and Intermediate Mass Ratio Inspirals - are unique probes of the astrophysical environments in which they evolve. Their long-lived and intricate dynamics allow for precise inference of source properties, provided waveform models are accurate enough to capture the full complexity of their orbital evolution. In this work, we develop a multi-parameter formalism, inspired by vacuum perturbation theory, to model asymmetric binaries embedded in general matter distributions with both radial and tangential pressures. In the regime of small deviations from the Schwarzschild metric, relevant to most astrophysical scenarios, the system admits a simplified description, where both metric and fluid perturbations can be cast into wave equations closely related to those of the vacuum case. This framework offers a practical approach to modeling the dynamics and the gravitational wave emission from binaries in realistic matter distributions, and can be modularly integrated with existing results for vacuum sources.

We study expansive measures for continuous flows without fixed points on compact metric spaces. We provide a new characterization of expansive measures through dynamical balls that, in contrast to the dynamical balls considered in [\emph{J. Differ. Equ.}, 256 (2014):2246--2260], are actually Borel sets. This makes the theory more amenable to measure-theoretic analysis. We prove that every ergodic invariant measure with positive entropy is positively expansive, extending the results of \emph{Ergod. Th. \& Dynam. Sys.} \textbf{4}(3) (2014):765--776] to the setting of flows. This implies that flows with positive topological entropy admit expansive invariant measures. Furthermore, we show that the stable classes of such measures have zero measure. Lastly, we prove that the set of expansive measures for a flow is a $G_{δσ}$-subset of the space of all probability measures and that every expansive measure (invariant or not) can be approximated by expansive measures supported on invariant sets.