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A variety of scientific fields like proteomics and spintronics have created a new demand for on-chip devices capable of sensing parameters localized within a few tens of micrometers. Nano and microelectromechanical systems (NEMS/MEMS) are extensively employed for monitoring parameters that exert uniform forces over hundreds of micrometers or more, such as acceleration, pressure, and magnetic fields. However, they can show significantly degraded sensing performance when targeting more localized parameters, like the mass of a single cell. To address this challenge, we present a new MEMS device that leverages the destructive interference of two topological radiofrequency (RF) counter-propagating wave modes along a piezoelectric Aluminum Scandium Nitride (AlScN) Su-Schrieffer-Heeger (SSH) interface. The reported MEMS device opens up opportunities for further purposes, including achieving more stable frequency sources for communication and timing applications.

For the Laplacian of an $n$-Riemannian manifold $X$, the Weyl law states that the $k$-th eigenvalue is asymptotically proportional to $(k/V)^{2/n}$, where $V$ is the volume of $X$. We show that this result can be derived via physical considerations by demanding that the gravitational potential for a compactification on $X$ behaves in the expected $(4+n)$-dimensional way at short distances. In simple product compactifications, when particle motion on $X$ is ergodic, for large $k$ the eigenfunctions oscillate around a constant, and the argument is relatively straightforward. The Weyl law thus allows to reconstruct the four-dimensional Planck mass from the asymptotics of the masses of the spin 2 Kaluza--Klein modes. For warped compactifications, a puzzle appears: the Weyl law still depends on the ordinary volume $V$, while the Planck mass famously depends on a weighted volume obtained as an integral of the warping function. We resolve this tension by arguing that in the ergodic case the eigenfunctions oscillate now around a power of the warping function rather than around a constant, a property that we call weighted quantum ergodicity. This has implications for the problem of gravity localization, which we discuss. We show that for spaces with D$p$-brane singularities the spectrum is discrete only for $p =6,7,8$, and for these cases we rigorously prove the Weyl law by applying modern techniques from RCD theory., Comment: 35 pages, 1 figure

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a lower bound on the size of the nodal set of a Laplacian eigenfunction, and different estimates involving the Wasserstein distance. The approach works in large variety settings, including Riemannian manifolds with a variable Kato-type lower bound on the Ricci curvature tensor, $\mathsf{RCD}(K,\infty)$ spaces, and some sub-Riemannian structures, such as Carnot groups, the Grushin plane and the $\mathbb{SU}(2)$ group., Comment: 23 pages

We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is related to the behavior of the potential functions $q_1$ and $q_2$ and to the asymptotic geometry of the end. The main ingredient is a new Carleman estimate of independent interest. Geometric applications to conformal deformations and to minimal graphs are presented., Comment: Minor corrections following referee's suggestions. Final version, to appear on IMRN

We prove the Pleijel theorem in non-collapsed RCD spaces, providing an asymptotic upper bound on the number of nodal domains of Laplacian eigenfunctions. As a consequence, we obtain that the Courant nodal domain theorem holds except at most for a finite number of eigenvalues. More in general, we show that the same result is valid for Neumann (resp. Dirichlet) eigenfunctions on uniform domains (resp. bounded open sets). This is new even in the Euclidean space, where the Pleijel theorem in the Neumann case was open under low boundary-regularity., Comment: Added the Dirichlet case and fixed minor typos

Inspired by recent graded metamaterials designs, we create phononic arrays of micro-resonators for frequency signal amplification and wave filtering. Leveraging suspended waveguides on a thick silicon substrate, we hybridize surface Rayleigh and Lamb flexural waves to effectively achieve phononic signal control along predefined channels. The guided waves are then spatially controlled using a suitable grading of the micro-resonators, which provide high signal-to-noise ratio and simultaneously create phononic delay-lines. The proposed device can be used for sensing, wave filtering or energy harvesting.

In models with extra dimensions, matter particles can be easily localized to a 'brane world', but gravitational attraction tends to spread out in the extra dimensions unless they are small. Strong warping gradients can help localize gravity closer to the brane. In this note we give a mathematically rigorous proof that the internal wave-function of the massless graviton is constant as an eigenfunction of the weighted Laplacian, and hence is a power of the warping as a bound state in an analogue Schr\"odinger potential. This holds even in presence of singularities induced by thin branes. We also reassess the status of AdS vacuum solutions where the graviton is massive. We prove a bound on scale separation for such models, as an application of our recent results on KK masses. We also use them to estimate the scale at which gravity is localized, without having to compute the spectrum explicitly. For example, we point out that localization can be obtained at least up to the cosmological scale in string/M-theory solutions with infinite-volume Riemann surfaces; and in a known class of N = 4 models, when the number of NS5- and D5-branes is roughly equal., Comment: 43 pages, 2 figures

Elastic metamaterials made from locally resonant arrays have been developed as effective ways to create band gaps for elastic or acoustic travelling waves. They work by implementing stationary states in the structure that localise and partially reflect waves. A different, simpler, way of obtaining band gaps is using phononic crystals, where the generated band gaps come from the periodic reflection and phase cancellation of travelling waves. In this work a different metamaterial structure that generates band gaps by means of coupling two contra-propagating modes is reported. This metamaterial, as it will be shown numerically and experimentally, generates larger band gaps with lower added mass, providing benefits for lighter structures.

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant $\Lambda$; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of $\Lambda$ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy., Comment: 57 pages + 4 appendices, 4 figures. v2: expanded introduction and comments on the Casimir vacuum. v3: clarification on the agreement of the Casimir vacuum with conjectured bounds

A reinforcement learning approach to design optimised graded metamaterials for mechanical energy confinement and amplification is described. Through the proximal policy optimisation algorithm, the reinforcement agent is trained to optimally set the lengths and the spacing of an array of resonators. The design optimisation problem is formalised in a Markov decision problem by splitting the optimisation procedure into a discrete number of decisions. Being the physics of graded metamaterials governed by the spatial distribution of local resonances, the space of possible configurations is constrained by using a continuous function for the resonators arrangement. A preliminary analytical investigation has been performed to characterise the dispersive properties of the analysed system by treating it as a locally resonant system. The outcomes of the optimisation procedure confirms the results of previous investigations, highlighting both the validity of the proposed approach and the robustness of the systems of graded resonators when employed for mechanical energy confinement and amplification. The role of the resonator spacing is shown to be secondary with respect to the resonator lengths or, in other words, with respect to the oscillation frequencies of the resonators. However, it is also demonstrated that reducing the number of resonators can be advantageous. The outcomes related to the joint optimisation of the resonator lengths and spacing, thanks also to the adaptive control of the analysis duration, overcome significantly the performance of previously known systems by working almost uniquely on enlarging the time in which the harvester oscillations take place without amplifying these oscillations. The proposed procedure is suitable to be applied to a wide range of design optimisation problems in which the effect of the design choices can be assessed through numerical simulations., Comment: 41 pages

While roton dispersion relations had been restricted to correlated quantum systems at low temperature, recent works show the possibility of obtaining this unusual dispersion in acoustic and elastic metamaterials. Such phenomenon has been demonstrated in periodic structures by means of beyond-nearest-neighbor interactions, following the formulation firstly developed by Brillouin in the $'50s$. In this paper, we demonstrate both numerically and experimentally that beyond-nearest-neighbor connections are not a necessary condition to obtain this unusual dispersion relation in elasticity. Leveraging the intrinsic complexity of elastic systems supporting different types of waves, we demonstrate that mode locking can be applied to obtain roton dispersion, without the need of elastic or magnetic interactions between non nearest neighbors. Moreover, the combination of roton dispersion and rainbow physics enables spatial separation of the energy fluxes with positive and negative group velocity.

Precise manipulation of the direction and re-direction of vibrational wave energy is a key demand in wave physics and engineering. We consider the paradigm of a finite frame-like structure and the requirement to channel energy away from critical regions, leaving them vibration-free, and redirect energy along edges towards energy concentrators for damping or energy harvesting. We design an exemplar frame metamaterial, combining two distinct areas of wave physics. Firstly, topological edge states taking an unconventional tetrachiral lattice. We control these highly localised protected edge states leveraging a hierarchy of scales through the addition of micro-resonators that impose tuneable symmetry breaking and reconfigurable mass. This allows us to achieve precise positional control in the macro-scale frame lattice, thereby opening opportunities for robust signal transport and vibration control. Experiments, theory, simulation are all utilised to provide a comprehensive analysis and interpretation of the physics.

We derive analytical solutions based on singular Green's functions, which enable efficient computations of scattering simulations or Floquet-Bloch dispersion relations for waves propagating through an elastic plate, whose surface is patterned by periodic arrays of elastic beams. Our methodology is versatile and allows us to solve a range of problems regarding arrangements of multiple beams per primitive cell, over Bragg to deep-subwavelength scales; we cross-verify against finite element numerical simulations to gain further confidence in our approach, which relies upon the hypothesis of Euler-Bernoulli beam theory considerably simplifying continuity conditions such that each beam can be replaced by point forces and moments applied to the neutral plane of the plate. The representations of Green's functions by Fourier series or Fourier transforms readily follows, yielding rapid and accurate analytical schemes. The accuracy and flexibility of our solutions are demonstrated by engineering topologically non-trivial states, from primitive cells with broken spatial symmetries, following the phononic analogue of the Quantum Valley Hall Effect (QVHE). Topologically protected states are produced and coexist along: interfaces between adjoining chiral-mirrored bulk media and edges between one such chiral bulk and the surrounding bare elastic plate, allowing topological circuits to be designed with robust waveguiding; these topologically non-trivial states exist within near flexural resonances of the constituent beams of the phononic crystal, and hence can be tuned into a deep-subwavelength regime.

This work studies a broadband graded metamaterial, which integrates the piezoelectric energy harvesting function targeting low-frequency structural vibrations, lying below 100 Hz. The device combines a graded metamaterial with beam-like resonators, piezoelectric patches and a self-powered piezoelectric interface circuit for energy harvesting. Based on the mechanical and electrical lumped parameters, an integrated model is proposed to investigate the power performance of the proposed design. Thorough numerical simulations were conducted to analyse the spatial frequency separation capacity and the slow-wave phenomenon of the graded metamaterial for broadband and high-capability piezoelectric energy harvesting. Experiments with realistic vibration sources show that the harvested power of the proposed design yields a five-fold increase with respect to conventional harvesting solutions based on single cantilever harvesters. Our results reveal the significant potential on exploitation of graded metamaterials for energy-efficient vibration-powered devices.

Research into the orbital angular momentum carried by helical wave-fronts has been dominated by the fields of electromagnetism and acoustics, owing to its practical utility in sensing, communication and tweezing. Despite the huge research effort across the wave community, only recently has elastic orbital angular momentum been theoretically shown to exhibit similar properties. Here we experimentally observe the transfer of elastic orbital angular momentum from a hollow elastic pipe to a fluid in which the pipe is partially submerged, in an elastic analogue of Durnin's slit-ring experiment for optical beams. This transfer is achieved by coupling the dilatational component of guided flexural waves in the pipe with the pressure field in the fluid; the circumferential distribution of the normal stress in the pipe acts as a continuous phased pressure source in the fluid resulting in the generation of Bessel-like acoustic beams. This demonstration has implications for future research into a new regime of orbital angular momentum for elastic waves, as well providing a new method to excite acoustic beams that carry orbital angular momentum that could create a new paradigm shift for acoustic tweezing., Comment: 11 Pages, 5 figures

We experimentally achieve selective wave filtering and polarization control in a three-dimensional elastic frame embedding local resonators. By connecting multi-resonating elements to a frame structure, a complete low-frequency, subwavelength bandgap with strong selective filtering properties is obtained. Theory and experiments demonstrate the metaframe capability to selectively stop transverse waves while allowing longitudinal wave propagation, as in 'fluid-like' elasticity. This peculiar behaviour, together with the complete bandgap structure, may open opportunities in the context of wave control, envisaging concurrent applications for three-dimensional filters and elastic wave polarizers.

We identify that flexural guided elastic waves in elastic pipes carry a well-defined orbital angular momentum associated with the compressional dilatational potential. This enables the transfer of elastic orbital angular momentum, that we numerically demonstrate, through the coupling of the compressional potential in a pipe to the acoustic pressure field in a surrounding fluid in contact with the pipe., Comment: 5 pages, 3 figures plus supplemental material

We design a device for the passive mode conversion of guided, axisymmetric, ultrasonic waves in hollow elastic pipes into arbitrary non-axisymmetric flexural waves that have a constant angular profile along the pipe axis. To achieve this we create an elastic analogue to optical spiral phase plates - the elastic spiral phase pipe. Three possible configurations of the elastic spiral phase pipe are presented which allow the efficient generation of non-axisymmetric flexural waves from an axisymmetric, longitudinal forcing. The theory leverages the dispersive nature of the guided elastic waves that are supported in pipes through a defined relative refractive index. As such we include a spectral collocation method used to aid the design of the elastic spiral phase pipe that is corroborated with numerical simulations and then experimentally verified., Comment: 34 pages, 10 figures

We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry--\'Emery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdS$_d$ vacua with a bridge admitting an AdS$_{d+1}$ interpretation, the holographic dual is a CFT$_d$ with two CFT$_{d-1}$ boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for $d=4$. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it., Comment: 61 pages, 4 figures

We experimentally achieve wave mode conversion and rainbow trapping in an elastic waveguide loaded with an array of resonators. Rainbow trapping is a phenomenon that induces wave confinement as a result of a spatial variation of the wave velocity, here promoted by gently varying the length of consecutive resonators. By breaking the geometrical symmetry of the waveguide, we combine the wave speed reduction with a reflection mechanism that mode-converts flexural waves impinging on the array into torsional waves travelling along opposite directions. The framework presented herein may open opportunities in the context of wave manipulation through the realization of structural components with concurrent wave conversion and energy trapping capabilities.

In the paper we prove two inequalities in the setting of ${\sf RCD}(K,\infty)$ spaces using similar techniques. The first one is an indeterminacy estimate involving the $p$-Wasserstein distance between the positive part and the negative part of an $L^{\infty}$ function and the measure of the interface between the positive part and the negative part. The second one is a conjectured lower bound on the $p$-Wasserstein distance between the positive and negative parts of a Laplace eigenfunction., Comment: Major update. Structure of the proof simplified. Added new results on indeterminacy estimates for transport distances. The inequality between $p$-Wasserstein and $p$-Hellinger distances for $p>2$ is now not needed: it has been removed and will be part of a future project

Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" $\mathbf{D}$-distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense., Comment: 39 pages. Added a section with a comparison with the conic Gromov-Wasserstein distance. Some other minor improvements. Final version, to appear on "Probability Theory and Related Fields"

We prove that the sharp Buser's inequality obtained in the framework of $\mathsf{RCD}(1,\infty)$ spaces by the first two authors is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of $\mathsf{RCD}(K,\infty)$ spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained., Comment: Added new results: the discussion on Cheeger's inequality now fits into the study of a family of inequalities relating eigenvalues of the p-Laplacian. To appear on Journal of Functional Analysis

We amalgamate two fundamental designs from distinct areas of wave control in physics, and place them in the setting of elasticity. Graded elastic metasurfaces, so-called metawedges, are combined with the now classical Su-Schrieffer-Heeger (SSH) model from the field of topological insulators. The resulting structures form one-dimensional graded-SSH-metawedges that support multiple, simultaneous, topologically protected edge states. These robust, enhanced localised modes are leveraged for applications in elastic energy harvesting using the piezoelectric effect. The designs we develop are first motivated by applying the SSH model to mass-loaded Kirchhoff-Love thin elastic plates. We then extend these ideas to using graded resonant rods, and create SSH models, coupled to elastic beams and full elastic half-spaces.