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Analytic representation formulas and power series are developed to describe the band structure inside periodic elastic crystals made from high contrast inclusions. We use source free modes associated with structural spectra to represent the solution operator of the Lamé system inside phononic crystals. Convergent power series for the Bloch wave spectrum are obtained using the representation formulas. An explicit bound on the convergence radius is given through the structural spectra of the inclusion array and the Dirichlet spectra of the inclusions. Sufficient conditions for the separation of spectral branches of the dispersion relation for any fixed quasi-momentum are identified. A condition is found that is sufficient for the emergence of band gaps. [ABSTRACT FROM AUTHOR]

Aiming at accurately predicting the global Damping Loss Factor (DLF) for Highly Dissipative Structures (HDS), the current study uses the Wave Finite Element (WFE) methodology. It starts by deriving the forced responses of a Unit Cell (UC) representative of the periodic meta-structure. Then it computes the DLF of the wave via the power balance. The Bloch expansion is employed. The response to a point force applied to the periodic structure is decomposed in the Brillouin zone, allowing the prediction via integration over the wavespace. The global DLF is derived based on the Power Input Method (PIM). The accuracy of the methodology is demonstrated through several cases from simple panels to complex meta-structures. For HDS, results of General Laminated Model (GLM) is exploited for wave DLF and PIM based on Finite Element Method (FEM) data is provided as reference approach for global DLF. The study discusses the influence of bending waves on the DLF estimation for HDS. A final case study with a meta-structure is also offered. The later consists of a doubly periodic coated sphere in a host rubber, it demonstrates the importance of Bloch modes. • Wave-based methodology is developed by Bloch wave expansion to estimate DLF of HDS. • DLF computation confirms the reduced impact of bending wave on HDS forced response. • Higher Bloch modes are essential for DLF estimation of meta-structure with large UC. [ABSTRACT FROM AUTHOR]

The dynamics of rotating systems plays an important role in the efficiency and reliability of such machines. For this reason, understanding their dynamics is essential to predict and mitigate the resultant vibrations that can occur in such machines. Considering that vibration is essentially the propagation of waves in a continuous medium, methods of wave analysis can also be employed to investigate the dynamics of rotating systems, especially when the system presents some sort of periodicity. The difficulty lies in the fact that rotating systems are non-reciprocal systems due to the gyroscopic effects, which create forward and backward whirl motions with associated mode waves. The conventional wave methods usually do not tackle this problem. This work presents a methodology for analyzing rotating systems with wave methods based on the combination of conventional modal analysis, Bloch wave propagation theory, and directional frequency response functions. In this case, it is possible to separate the forward and backward modes, thus allowing the analysis of forward and backward wavenumbers separately. Consequently, one can draw separate dispersion diagrams for the wave modes associated with the forward and backward motions of the rotor. The numerical results show the feasibility of the method, and the methodology is applied experimentally to a rotating system in the laboratory. [Display omitted] • Wave analysis of a periodic rotor considering gyroscopic effects. • Forward and backward whirl modes belong to different wave groups. • Band gaps can differ from non-propagating and strongly attenuated propagation. • Wave Campbell diagram separates the modes and show the respective wavenumbers. [ABSTRACT FROM AUTHOR]

The overexpression and/or amplification of the HER2/neu oncogene has been proposed as a prognostic marker in breast cancer. The detection of the related peptide HER2 remains a grand challenge in cancer diagnosis and for therapeutic decision-making. Here, we used a biosensing device based on Bloch Surface Waves excited on a one-dimensional photonic crystal (1DPC) as valid alternative to standard techniques. The 1DPC was optimized to operate in the visible spectrum and the biosensor optics has been designed to combine label-free and fluorescence operation modes. This feature enables a real-time monitoring of a direct competitive assay using detection mAbs conjugated with quantum dots for an accurate discrimination in fluorescence mode between HER2-positive/negative human plasma samples. Such a competitive assay was implemented using patterned alternating areas where HER2-Fc chimera and reference molecules were bio-conjugated and monitored in a multiplexed way. By combining Label-Free and fluorescence detection analysis, we were able to tune the parameters of the assay and provide an HER2 detection in human plasma in less than 20 min, allowing for a cost-effective assay and rapid turnaround time. The proposed approach offers a promising technique capable of performing combined label-free and fluorescence detection for both diagnosis and therapeutic monitoring of diseases. [Display omitted] • Direct detection of HER2 peptide for monitoring the overexpression of the HER2/neu oncogene, marker of the breast cancer. • Competitive assay for an accurate discrimination in fluorescence mode in less than 20 min between HER2-positive/negative human plasma samples. • The proposed platform combines label-free and fluorescence detection for both diagnosis and therapeutic monitoring of diseases. • Low-cost disposable biochips sustaining Bloch surface waves in the visible range on 1D-photonic crystals deposited on plastic substrates. [ABSTRACT FROM AUTHOR]

We study wave-packets in nonlinear periodic media in arbitrary (d) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad wave-packets with N ∈ N carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an L 1 -norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the d -dimensional CMEs, we discuss spectral gaps of the CMEs. For N = 4 and d = 2 a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for d > 1 and for the considered values of N due to the absence of a spectral gap in the standard moving frame variables. [ABSTRACT FROM AUTHOR]

We study the transformation properties of the electron states in crystals with spin–orbit coupling, focusing primarily on the limitations of the frequently used pseudospin- 1 ∕ 2 description of twofold degenerate Bloch bands. Using the language of corepresentations of magnetic point groups, we construct the Bloch bases across the Brillouin zone in a way which is consistent with all symmetry requirements. This construction is applied to derive the effective spin–orbit Hamiltonians in noncentrosymmetric crystals, known as the generalized Rashba models, in both single-band and multiband cases. [ABSTRACT FROM AUTHOR]

Abstract We address the existence and stability properties of localized truncated-Bloch-wave solitons in one-dimensional nonlinear fractional Schrödinger equation with an optical lattice. Under a defocusing nonlinearity, extended nonlinear Bloch waves with finite amplitudes bifurcate from linear Bloch modes at band edges. The variation of Lévy index has a profound impact on the existence, profile, power and stability of nonlinear modes. Truncated-Bloch-wave solitons with different number of units are found in finite bandgaps. The main body of truncated-Bloch-wave soliton coincides exactly with the corresponding nonlinear Bloch wave. While solitons residing in the second gap propagate stably in a narrow region, nonlinear states in the first gap are stable in almost their whole existence domains. Specially, the instability of solitons can be significantly suppressed by the decrease of Lévy index. [ABSTRACT FROM AUTHOR]

Abstract The band structure, Bloch functions, density of states, Fermi velocity, and inter-miniband absorption coefficient have been calculated for a one-layered InAs/GaAs quantum dot superlattice of honeycomb symmetry, taking into account the elastic strain distribution because of the lattice mismatch between the materials of quantum dots and the matrix. Green's function formalism in combination with the Fourier transformation to reciprocal space is applied to obtain the miniband energy dispersions and Bloch functions via the exact diagonalization procedure. It is shown that the strain anisotropy leads to the shift of the Dirac points from the K and K ' points in the first Brillouin zone. As a result an anisotropy in the Fermi velocity as well as qualitative changes in the density of states and inter-miniband absorption coefficient have been observed. We believe that in addition to providing an information about the main characteristics of honeycomb superlattices, our study may also open up new possibilities to design novel devices based on artificial graphene. Highlights • Elastic strain distribution in a honeycomb superlattice has been obtained theoretically. • Miniband dispersion surfaces in conduction band have been obtained. • Density of states, Fermi velocity and absorption coefficient are calculated as well. [ABSTRACT FROM AUTHOR]

Abstract Waveguiding in a phononic crystal (PC) can be achieved along either linear line defects or a sequence of cavities, for frequencies belonging to a complete bandgap. When waves are coupled inside a PC waveguide, modulations in the frequency transmission are generally observed, leading to the formation of a channeled spectrum. We show that the channeled spectrum results from the interference of forward and backward guided Bloch waves. We first theoretically develop a Bloch wave interference model. Then, we consider the case of linear waveguides and of coupled-resonator waveguides formed in a 2D square PC composed of water cylinders in mercury. The transmission properties of waveguides with different length and defect distribution are examined. In all cases, the observed channeled spectra are well explained by the theoretical model. This work is relevant to the design of new acoustic and elastic wave devices. [ABSTRACT FROM AUTHOR]

Abstract The stop and pass bands (i.e. the band gaps) characteristics are determined for gyroscopic systems by developing an approach which is compatible with such class of systems which is based on the concept of Bloch wave propagation in periodic structures. In this approach, the dispersion curves of the periodic gyroscopic systems are determined for different rotational speeds. The obtained characteristics are compared with non-rotating systems in an attempt to quantify the effect of the gyroscopic forces on the “ band gap ” characteristics. The developed approach is illustrated by a new class of drill strings with passive periodic inserts. These inserts are utilized to filter out the vibration transmission along the drill string. Such mechanical filtering capabilities allow the vibrations to propagate along the periodic drill string only within specific frequency bands called the ‘ pass bands ’ and completely block it within other frequency bands called the ‘ stop bands ’. The inserts introduce impedance mismatch zones along the vibration transmission path to impede the propagation of vibration along the drill string. The design and the location of the inserts are selected to confine the dominant modes of vibration of the drill string within the stop bands generated by the periodic arrangement of the inserts in order to completely block the propagation of the vibrations. A finite element model (FEM) that simulates the operation of this new class of drill strings is developed to describe the complex nature of the vibration encountered during drilling operations. The FEM is used to extract the dispersion characteristics of the gyroscopic unit cell of the drill string in order to determine its stop and pass band characteristics. Experimental prototype of the passive periodic drill string is built and tested to demonstrate the feasibility and effectiveness of the concept of periodic drill string in mitigating undesirable vibrations. The experimental results are used to validate the developed theoretical model in order to develop a scalable design tool that can be used to predict the dynamical behavior of this new class of drill strings. [ABSTRACT FROM AUTHOR]

We investigate the complex band structure of elastic waves in a magnetoelectroelastic phononic crystal, consisting of a polymeric matrix reinforced by BaTiO 3 -CoFe 2 O 4 inclusions in a square lattice. We also study the influence of inclusion geometry - circular, square and rotated square on the band structure. The elastic wave propagation inside periodic structures consists of both propagating and evanescent modes. Accordingly, improved plane wave expansion ( ω ( k ) ) and extended plane wave expansion ( k ( ω ) ) methods are formulated to solve the governing equations of motion of a magnetoelectroelastic solid considering wave propagation in the xy plane in order to obtain the propagating and evanescent modes. Complete band gaps between the XY and Z modes are observed for all types of inclusion. Piezoelectricity and piezomagnetism influence significantly the band gaps and the attenuation. There are many modes of the magnetoelectroelastic phononic crystal that are not identified by the ω ( k ) method. However, they are well identified by the k ( ω ) method proposed. These complex modes are related to the evanescent nature of Bloch waves. The new findings of this study should be useful for vibration control using smart periodic structures with piezoelectricity and piezomagnetism. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball B n in R n ( n ≥ 2 ) . Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space B ( B n ) and the generalized Bloch space L ∞ , ω B α , a 0 ( B n ) , respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric. [ABSTRACT FROM AUTHOR]

Meta-structures with dynamic vibrational resonators (DVRs) are programmed to control the propagation of waves and attenuate vibrations over a broadband frequency spectrum. Attributes of DVRs, such as their resonant frequency and mass, determine the location and width of the bandgap, respectively. As a result, to adaptively program bandgaps, one has to modify or tune the eigenvalues of individual DVRs, and a popular approach is to vary the stiffness of each resonator. However, the tunable range of bandgaps is often restricted to maximum change in DVRs' stiffness. This work presents a novel approach to adaptively program bandgaps of a 1D flexural meta-structure over a broad frequency bandwidth. DVRs with two stable configurations are attached to a beam in developing the meta-structure. A numerical model is developed to illustrate the scope of the novel approach. An experimental investigation then validates the simulated results and shows the extent of the vibration absorption capabilities of the meta-structure. A reinforced learning approach is used to adaptively tune the bandgap over 220 Hz to 840 Hz. • Various configurations of meta-structures are introduced with bistable resonators. • Multiple bandgap formation is explained by introducing patterns of n-bit unitcells. • Bandgap locations for various unitcell patterns is simulated by Bloch wave analysis. • Validation of Finite element formulation is conducted by experiments. • RL is implemented to switch between different configurations of meta-structure. [ABSTRACT FROM AUTHOR]

The Bloch mode synthesis (BMS) model-reduction method adapts component mode synthesis techniques to unit-cell problems in order to obtain a reduced-order model that quickly produces band-structure frequencies for any wave vector, or vice versa. Fundamental to BMS is a partitioning of the real-space model into interior and boundary components, and subsequent reduction of the interior via truncated normal mode expansion. In this paper, two enhancements are presented for the BMS method that reduce both computation time and error in band-structure calculations. The first enhancement improves the accuracy of the interior reduction by approximating the participation of the residual modes rather than simply truncating them. The original formulation of BMS includes a modal reduction of the boundary that must be recomputed for every wave vector. This limits computational benefits and prevents the reduced-order model from being useful for the inverse band-structure problem (i.e., the k ( ω ) calculation). The second enhancement is a local boundary reduction that is independent of wave vector and thus does not suffer from the aforementioned limitations. [ABSTRACT FROM AUTHOR]

The microwave magnetic field strength, B 1 , in the cavity of a conventional continuous wave electron paramagnetic resonance, CW-EPR, spectrometer was measured by employing modulation sidebands, MS, in the EPR spectrum. MS spectrum in CW-EPR is produced by applying the modulation frequency, ω rf , which exceeds the linewidth, δB , given in frequency units. An amplitude-modulated CW-EPR, AM-CW-EPR, was selected as detection method. Theoretical description of AM-CW-EPR spectrum was modified by adding Bloch-Siegert-like shift obtained by taking into account the cumulative effect of the non-resonant interactions between the driving fields and the spin system. This approach enables to enhance the precision of B 1 measurement. In order to increase the sensitivity of the method when saturation effects, due to higher intensity of B 1 , decrease the resolution of AM-CW-EPR spectrum, detection at the second harmonic of CW-EPR has been employed. [ABSTRACT FROM AUTHOR]

We study locally univalent functions f analytic in the unit disc D of the complex plane such that | f ″ ( z ) / f ′ ( z ) | ( 1 − | z | 2 ) ≤ 1 + C ( 1 − | z | ) holds for all z ∈ D , for some C ∈ ( 0 , ∞ ) . If C ≤ 1 , then f is univalent by Becker's univalence criterion. We discover that for C ∈ ( 1 , ∞ ) the function f remains to be univalent in certain horodiscs. Sufficient conditions which imply that f is bounded, belongs to the Bloch space or belongs to the class of normal functions, are discussed. Moreover, we consider generalizations for locally univalent harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper we give some characterizations of holomorphic Campanato type spaces in terms of Carleson tubes and Bergman metric balls in the unit ball. [ABSTRACT FROM AUTHOR]

The objective of this paper is to use transfer functions to comprehend the formation of band gaps in locally resonant acoustic metamaterials. Identifying a recursive approach for any number of serially arranged locally resonant mass in mass cells, a closed form expression for the transfer function is derived. Analysis of the end-to-end transfer function helps identify the fundamental mechanism for the band gap formation in a finite metamaterial. This mechanism includes (a) repeated complex conjugate zeros located at the natural frequency of the individual local resonators, (b) the presence of two poles which flank the band gap, and (c) the absence of poles in the band-gap. Analysis of the finite cell dynamics are compared to the Bloch-wave analysis of infinitely long metamaterials to confirm the theoretical limits of the band gap estimated by the transfer function modeling. The analysis also explains how the band gap evolves as the number of cells in the metamaterial chain increases and highlights how the response varies depending on the chosen sensing location along the length of the metamaterial. The proposed transfer function approach to compute and evaluate band gaps in locally resonant structures provides a framework for the exploitation of control techniques to modify and tune band gaps in finite metamaterial realizations. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for the operator of integration to be bounded on H ∞ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. [ABSTRACT FROM AUTHOR]

Let P denote the Bergman projection on the unit disk D , P μ ( z ) : = ∫ D μ ( w ) ( 1 − z w ¯ ) 2 d A ( w ) , z ∈ D , where d A is normalized area measure. We prove that if | μ ( z ) | ≤ 1 on D , then the integral I μ ( a , r ) : = ∫ 0 2 π exp ⁡ { a r 4 | P μ ( r e i θ ) | 2 log ⁡ 1 1 − r 2 } d θ 2 π , 0 < r < 1 , has the bound I μ ( a , r ) ≤ C ( a ) : = 10 ( 1 − a ) − 3 / 2 for 0 < a < 1 , irrespective of the choice of the function μ . Moreover, for a > 1 , no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection P μ (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0 < a < π 2 64 = 0.1542 …  . We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k -quasiconformal extension, for 0 < k < 1 . The estimate reads, for t ∈ C and 0 < k < 1 , B ( k , t ) ≤ { 1 4 k 2 | t | 2 ( 1 + 7 k ) 2 , for | t | ≤ 2 k ( 1 + 7 k ) 2 , k | t | − 1 ( 1 + 7 k ) 2 , for | t | ≥ 2 k ( 1 + 7 k ) 2 , which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with | t | ≤ 2 / k , we should have B ( k , t ) = 1 4 k 2 t 2 . [ABSTRACT FROM AUTHOR]

We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schrödinger/Gross–Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinear Schrödinger equation. Besides a formal derivation of the CMEs a rigorous justification of the approximation and an error estimate in the supremum norm are provided. Several numerical tests corroborate the analysis. [ABSTRACT FROM AUTHOR]

We introduce and characterize Bloch functions on bounded symmetric domains, which may be infinite dimensional, by extending several well-known equivalent conditions for Bloch functions on the open unit disc U in C . We also generalize a number of results concerning Bloch functions on U to bounded symmetric domains. Given a holomorphic mapping φ between bounded symmetric domains B X and B Y , we derive criteria for boundedness and compactness of the composition operator C φ between the Bloch spaces B ( B Y ) and B ( B X ) , extending several known results for finite dimensional domains. [ABSTRACT FROM AUTHOR]

• A theoretical model is proposed for Bloch waves in a soft lattice. • The coupled modes are converted into easy modes by finite stretching. • Mode evolution is governed by effective shear modulus and length of the component. • The broadest directional bandgap is observed for equi-biaxial stretch. • Theoretically predicted flat bands do not exist in real soft lattices. In this study, the in-plane Bloch wave propagation in a finitely stretched square lattice is investigated numerically and theoretically. Precisely, the elastic band diagram is calculated for an infinite periodic structure with a cruciform hyperelastic unit cell under uniaxial or biaxial tension. Meanwhile, an elastodynamic tight-binding model is proposed to investigate the formation and evolution of the band structure. In said lattice, the finite stretch was found to not only tune the symmetry of the band structure but also purify the elastic waves into easy modes. The uniaxial stretch exhibits the opposite impacts on the evolution of two easy modes, while the effect of the biaxial stretch is equated with the superposition of the uniaxial stretches in the two tessellation directions. The mentioned effects on the band structure could be attributed to the competition between the effective shear moduli and lengths for different beam components. As a result, the finite stretch could tune the directional elastic wave bandgap of the soft lattice, and the broadest bandgap could be anticipated in an equi-biaxial stretch. This study may shed some light on the design and implementation of elastic wave control devices with weight efficiency and tunability. It also reveals that the theoretically predicted flat bands did not exist in the numerical calculations, a difference between the physical system and the corresponding simplified theoretical model. [ABSTRACT FROM AUTHOR]

Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Attempts to develop the hyperbolic analogue of Bloch's theorem have revealed an intrinsic role for algebro-geometric moduli spaces, notably those of stable bundles on a curve. We expand this picture to include Higgs bundles, which enjoy natural interpretations in the context of band theory. First, their spectral data encodes a crystal lattice and momentum, providing a framework for symmetric hyperbolic crystals. Second, they act as a complex analogue of crystal momentum. As an application, we elicit a new perspective on Euclidean band theory. Finally, we speculate on potential interactions of hyperbolic band theory, facilitated by Higgs bundles, with other themes in mathematics and physics. [ABSTRACT FROM AUTHOR]

This paper presents a thermoacoustic model in the frequency domain for multi-burner combustors composed of multiple plenum-nozzle-chamber modules with cross-talk at both plenums and chambers. The thermoacoustic model is given as a multi-input-multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. The discrete rotational symmetry of a multi-burner combustor is exploited to obtain a simple, diagonal transfer function matrix. The resonance equation, whose roots are resonances, is given as a set of scalar equations. The explicit form of the resonance equation reveals how combustor and operating parameters influence the resonances, and which parameters are significant for the longitudinal-azimuthal coupling and the plenum-chamber coupling. The diagonalization also allows us to show constructively that every mode shape is a Bloch wave and those mode shapes can characterize all resonances in terms of longitudinal and azimuthal mode numbers. At a low frequency, the resonance equation can be approximated as biquadratic and quadratic equations for a typical combustor geometry and operating conditions. A bunch of low-frequency resonances, peculiar to multi-burner combustors with cross-talk, are interpreted as azimuthal Helmholtz modes in which plenums/chambers and cross-talk ducts serve as the cavities and necks, respectively. Our combustor model is combined with a diagonal flame transfer function matrix for combustion instability analysis and we develop a set of scalar stability equations whose roots determine combustion instability. This stability equation reveals how combustion instability depends on the combustor geometry and operating parameters. Several numerical examples are presented to validate our thermoacoustic model and theoretical findings. • Acoustically, a multi-burner combustor is a collection of multiple single-burner combustors with cross-talks. • Thermoacoustics of a multi-burner combustor is a multi-input-multi-output transfer function matrix. • Cross-talk gives rise to resonance clusters whose spread is prominent at low frequency. • Low-frequency resonances are Helmholtz modes where cross-talk ducts and nozzle serve as necks. • A chunky cross-talk duct gives a strong cross-talk effect at low frequency. [ABSTRACT FROM AUTHOR]

Abstract: A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrödinger equations in periodic media. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the WKB approximation and homogenization leads to the Bloch eigenvalue problem and an associated Hamilton–Jacobi system for the phase in each Bloch band, with the Bloch eigenvalue be part of the Hamiltonian. We formulate a level set description to capture multi-valued solutions to the band WKB system, and then evaluate total homogenized density over a sample set of bands. A superposition of band densities is established over all bands and solution branches when away from caustic points. The numerical approach splits the solution process into several parts: (i) initialize the level set function from the band decomposition of the initial data; (ii) solve the Bloch eigenvalue problem to compute Bloch waves; (iii) evolve the band level set equation to compute multi-valued velocity and density on each Bloch band; (iv) evaluate the total position density over a sample set of bands using Bloch waves and band densities obtained in steps (ii) and (iii), respectively. Numerical examples with different number of bands are shown to demonstrate the capacity of the method. [Copyright &y& Elsevier]

• Band structure of elastically connected periodic beams system is determined. • The unit cells are organized like the diatomic and triatomic chain. • Frequency response functions are used to validate the obtained dispersion curves. • Effects of medium stiffness and point masses on the band structure are investigated. This paper proposes the methodology to carry out the analysis of Bloch wave propagation in an array of vertically aligned and elastically connected structural elements such as beams, strings, plates, or other slender structures. The suggested approach is based on the Galerkin approximation and Floquet-Bloch theorem used in defining the eigenvalue problem and obtaining the band structure of the periodic systems. Special attention is devoted to the case of elastically connected Rayleigh beams with attached concentrated masses and wave propagation in the direction normal to the beam's length. A validation study is performed by using the finite element model and the frequency response function to confirm the accuracy of the solution obtained via the Galerkin approximation. Two configurations of unit cells, having two and three elastically connected beams with different geometrical and material properties, are considered in the numerical study. The effects of various parameters are investigated to reveal their influence on the frequency band structure and emergence of the zero-frequency bandgap. The results of this study demonstrates the tunability properties of the proposed periodic systems due to changes in values of concentrated masses, stiffness of the coupling medium or boundary conditions on structural elements within the unit cell. [ABSTRACT FROM AUTHOR]

• Metastructure with simple construction yet excellent low-frequency STL. • Semi-analytical method based on fast retrieval of dynamic effective mass. • Revealing of mechanisms via Bloch wave dispersion and Bloch wave modes. • Parametric design guideline for achieving broadband high diffuse STL. • Experimental validation of parametric design guideline. Recent studies on plate-type metastructures have shown promising applications in low-frequency sound insulation, as they possess a low-frequency sound transmission loss (STL) that is considerably higher than that derived from the mass law. To design a plate-type metastructure for a sound insulation application, three problems are worth exploring: 1) how to predict efficiently the STL of the metastructure, 2) how to understand its unusual STL behavior, and 3) how to design its parameters to obtain a high STL at a target frequency and achieve a broad frequency band of high STL without increasing the mass. The purpose of this work is to address these problems by investigating a plate-type metastructure that is simply constructed using a periodic array of mass blocks attached to a thin host plate. For the first problem, a semi-analytical method is proposed for efficient prediction of normal, oblique, and diffuse STL based on fast retrieval of dynamic surface mass density using a considerably reduced structural finite element model and without having to include the acoustic field. This method is validated based on comparisons with an existing numerical method and reported experimental data. For the second problem, the mechanisms of unusual STL behavior are revealed by analyzing dynamic surface mass density as well as Bloch wave dispersion and Bloch modes. The third problem is addressed by conducting an elaborate parametric analysis that provides straightforward guidelines for adjusting structural parameters to tune the STL peak frequency as well as offering general guidelines for designing multiple parameters to achieve a broader frequency band of high diffuse STL with a given surface mass density. A design guideline for improving STL bandwidth is experimentally validated through measurements of two specimens designed and fabricated with identical surface mass densities. Because the considered plate-type metastructure is simple yet instructive, these investigations and findings are of fundamental and practical interest to the design of sound insulation metastructures. [ABSTRACT FROM AUTHOR]

In this article, we offer a fast calculation method for the dispersive properties of three-dimensional micro-structured solids with one-dimensional and two-dimensional translation symmetry. We review mathematical methods to obtain the complex dispersion diagram for such structures (i. e. the Bloch vector as a function of frequency). This can be done by solving a Bloch-Floquet eigenvalue problem which is non-linear in the Bloch vector. Such a problem inevitably calls for reduction methods as the required number of degrees of freedom of the unit cell increases. Therefore, an asymptotic-in-frequency technique is implemented in order to approximate the dynamic stiffness matrix of the unit cell. This is done by identifying and retaining the most significant nodal degrees-of-freedom, which are used to generate a unit cell "superelement". The accuracy of the Bloch vectors and corresponding eigenvectors associated with the reduced non-linear eigenvalue problem is demonstrated by direct comparison to full-size computations and shows excellent agreement combined with considerable computing time reduction and controllable limitations. • We review of dispersion calculation and reduction methods for analyzing periodic micro-structured solids. • We study an algorithmic search for a ''retained set'' of dynamically significant finite-element (FE) degrees-of-freedom • We define a reduced FE mass and stiffness matrices of unit cells (a "superelement") using the retained set. • We compare superelement-based complex dispersion diagrams of periodic media to full FE models. • We discuss the controllable accuracy and computational efficiency of the novel method. [ABSTRACT FROM AUTHOR]

Let M = { m j , k } j , k = 1 + ∞ be an infinite complex-valued matrix which acts contractively on ℓ 2. For the weighted short diagonal sums S M (l) : = ∑ j , k : j + k = l (l j k) 1 2 m j , k , we obtain the estimate ∑ l = 2 + ∞ s l l | S M (l) | 2 ≤ 2 s log ⁡ e 1 − s , 0 ≤ s < 1. Expressed more vaguely, | S M (l) | 2 ⪅ 2 holds in the sense of averages. Concerning the optimality of the above bound, a construction due to Zachary Chase shows that the statement does not hold if the number 2 is replaced by the smaller number 1.72. In the construction, M is a permutation matrix. We interpret our bound in terms of the correlation E Φ (z) Ψ (z) of two copies of a Gaussian analytic function with possibly intricate Gaussian correlation structure between them. The Gaussian analytic function we study arises in connection with the classical Dirichlet space, which is naturally Möbius invariant. The study of the correlations E Φ (z) Ψ (z) leads us to introduce a new space, the mock-Bloch space (or Blochish space), which is slightly bigger than the standard Bloch space. Our bound has an interpretation in terms of McMullen's asymptotic variance, originally considered for functions in the Bloch space. Finally, we show that the correlations E Φ (z) Ψ (w) may be expressed as Dirichlet symbols of contractions on L 2 (D) , and show that the Dirichlet symbols of Grunsky operators associated with univalent functions find a natural characterization in terms of a nonlinear wave equation. [ABSTRACT FROM AUTHOR]

• An elastic-electro-mechanical model is built based on the Kirchhoff plate theory and equivalent impedance method. • The elastic-electro-mechanical model is numerically solved by using the Bloch theorem and wave finite element method. • Inserting an parallel inductor can greatly improve the performance of low-frequency and broadband vibration energy harvesting. Structural vibrations usually exist in the form of low-frequency and broadband elastic waves, so cantilever-like harvesters are not appropriate due to space limitations and high quality factor. Piezoelectric metamaterial plate with local resonators (PMP-LR) has been explored to overcome it. However, how to model and analyze the whole energy harvesting system is still a challenge. In this paper, a self-powered synchronized charge extraction circuit is presented and connected to the PMP-LP as the interface circuit. An elastic-electro-mechanical model is built based on the Kirchhoff plate theory and equivalent impedance method, where equivalent impedance of the self-powered synchronized charge extraction circuit is first derived. Then the elastic-electro-mechanical model is numerically solved by using the Bloch theorem and wave finite element method. By numerical simulations, it is found that the synchronized charge extraction circuit has few effects on vibration bandgaps of the PMP-LR. While by inserting an inductor (L r) parallel with the clamped capacitor (C p) of the piezoelectric patch, we can see that a new dispersion curve is induced by the L r - C p electrical resonance and the inductor is beneficial for low-frequency and broadband vibration energy harvesting. In particular, the inductor can greatly improve the harvesting performance when the resonant frequency is equal to the excitation frequency. In the end, experiments are done and the results are consistent with the numerical ones. Excitingly, the output voltage amplitude of the piezoelectric patch is enlarged about 200% after using the resonant inductor. [ABSTRACT FROM AUTHOR]

Using a weak limit for the hopping integral in one direction in the Hofstadter model, we show that the fermion states in the gaps of the spectrum are determined within the Kitaev chain. The proposed approach allows us to study the behavior of Chern insulators (CI) in different classes of symmetry. We consider the Hofstadter model on the square and honeycomb lattices in the case of rational and irrational magnetic fluxes ϕ , and discuss the behavior of the Hall conductance at a weak magnetic field in a sample of finite size. We show that in the semiclassical limit at the center of the fermion spectrum, the Bloch states of fermions turn into chiral Majorana fermion liquid when the magnetic scale 1 ϕ is equal to the sample size N. We are talking about the dielectric–metal phase transition, which is determined by the behavior of the Landau levels in 2D fermion systems in a transverse magnetic field. When a magnetic scale, which determines the wave function of fermions, exceeds the size of the sample, a jump in the longitudinal conductance occurs. The wave function describes non-localized states of fermions, the sample becomes a conductor, the system changes from the dielectric state to the metallic one. It is shown, that at 1 ∕ ϕ > N the quantum Hall effect and the Landau levels are not realized, which makes possibility to study the behavior of CI in irrational magnetic fluxes. [ABSTRACT FROM AUTHOR]