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We construct an order-sharp theory for a double-porosity model in the full linear elasticity setup. Crucially, we uncover time and frequency dispersive properties of highly oscillatory elastic composites., Comment: 52 pages, 4 figures

Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^d,$ $d\ge2,$ converging to metric graphs in the limit of vanishing thickness parameter in the resonant case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to $\delta'$ type., Comment: 2 figures; continuation of arXiv 1808.03961; as accepted in Mathematics

We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate the links of this rich area, on the one hand, to functional frameworks developed by S. N. Naboko and his students, and on the other hand, to concrete applications of current interest in the physics and engineering communities., Comment: 60 pages, 2 figures; a survey of recent results in the area, see also arXiv:2010.13318, arXiv:1808.03961, arXiv:1703.06220, arXiv:1510.03364

We provide resolvent asymptotics as well as various operator-norm estimates for the system of linear partial differential equations describing the thin infinite elastic rod with material coefficients which periodically highly oscillate along the rod. The resolvent asymptotics is derived simultaneously with respect to the thickness of the rod and the period of material oscillations. These two parameters are taken to be of the same order. The analysis is carried out separately on two invariant subspaces pertaining to the out-of-line and in-line displacements, under some additional assumptions, as well as in the general case where these two sorts of displacements intertwine inseparably., Comment: 59 pages

Motivated by recent results concerning the asymptotic behaviour of differential operators with highly contrasting coefficients, which have involved effective descriptions involving generalised resolvents, we construct the functional model for a typical example of the latter. This provides a spectral representation for the generalised resolvent, which can be utilised for further analysis, in particular the construction of the scattering operator in related wave propagation setups., Comment: 24 pages, 1 figure

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. By combining stochastic techniques with operator theory, we link objects defined on an abstract probability space to their counterparts in the physical space. In particular, we show that the key characteristics of the limit operator can be obtained from a single "typical" realisation in the physical space. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of statistically relevant limiting spectrum, we describe the connection between these two sets., Comment: 84 pages, no figures. arXiv admin note: text overlap with arXiv:1203.1140 by other authors

In 1991, Vladimir Maz'ya, Serguei Nazarov and Boris Plamenevskij developed the theory of compound asymptotics for elliptic boundary value problems in singularly perturbed domains. They considered a harmonic function whose domain contains a small inclusion. We applied this technique in the analysis of a Nematic liquid crystal with a small colloidal inclusion. However, we realised that the Maz'ya, Nazarov and Plamenevskij did not consider the asymptotic analysis of Robin boundary conditions, which corresponded to weak anchoring in the context of liquid crystals. In this piece we shall derive an asymptotic approximation to a harmonic function, in a domain with a small circular inclusion of radius $\epsilon>0$, with inhomogenous Robin boundary conditions and a corresponding parameter $\kappa>0$. We shall then prove that the difference between the exact solution and the approximation is uniformly bounded and derive the rate as a function of $\epsilon$ and $\kappa$., Comment: 17 pages, 0 figures

Two novel of boron-rich selenides, orthorhombic B$_6$Se and rhombohedral B$_{12}$Se, have been recently synthesized at high pressure - high temperature conditions. Room-temperature compressibilities of these phases were studied in a diamond anvil cell using synchrotron powder X-ray diffraction. A fit of experimental p-V data by third-order Birch-Murnaghan equation of state yielded the bulk moduli of 155(2) GPa for B$_{12}$Se and 144(3) GPa for B$_6$Se. No pressure-induced phase transitions have been observed in the studied pressure range, i.e., up to 35 GPa.

We analyse the behaviour of thin composite plates whose material properties vary periodically in-plane and possess a high degree of contrast between the individual components. Starting from the equations of three-dimensional linear elasticity that describe soft inclusions embedded in a relatively stiff thin-plate matrix, we derive the corresponding asymptotically equivalent two-dimensional plate equations. Our approach is based on recent results concerning decomposition of deformations with bounded scaled symmetrised gradients. Using an operator-theoretic approach, we calculate the limit resolvent and analyse the associated limit spectrum and effective evolution equations. We obtain our results under various asymptotic relations between the size of the soft inclusions (equivalently, the period) and the plate thickness as well as under various scaling combinations between the contrast, spectrum, and time. In particular, we demonstrate significant qualitative differences between the asymptotic models obtained in different regimes.

Two boron-rich chalcogenides B12S and B12Se isostructural to {\alpha}-rhombohedral boron were synthesized by chemical reaction of the elements at high-pressure - high-temperature conditions. The crystal structures and stoichiometries of both compounds were confirmed by Rietveld refinement and elemental analysis. The experimental Raman spectra of B12S and B12Se were investigated for the first time. All observed Raman bands have been attributed to the theoretically calculated phonon modes, and the mode assignment has been performed.

Using a generalisation of the classical notion of Dirichlet-to-Neumann map and the related formulae for the resolvents of boundary-value problems, we analyse the asymptotic behaviour of solutions to a "transmission problem" for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit problem of "electrostatic" type for functions that are constant on the inclusion. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates., Comment: 15 pages, 1 figure. Continuation of: arXiv:1907.08144. As accepted by: Math. Notes

For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse the Maxwell system of equations of electromagnetism on $\varepsilon$-periodic sets $S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures $\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed 1-periodic measure $\mu,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. In the resent work we address the case of non-zero current density in the Maxwell system and complete the analysis of the general setup including non-constant permittivity and permeability coefficients., Comment: 25 pages

New boron-rich sulfide B6S and selenide B6Se have been discovered from high pressure - high temperature synthesis combined with ab initio evolutionary crystal structure prediction, and studied by synchrotron X-ray diffraction and Raman spectroscopy at ambient conditions. As it follows from Rietveld refinement of powder X-ray diffraction data, both chalcogenides have orthorhombic symmetry and belongs to Pmna space group. All experimentally observed Raman bands have been attributed to the theoretically calculated phonon modes, and the mode assignment has been performed. Prediction of mechanical properties (hardness and elastic moduli) of new boron-rich chalcogenides have been made using ab initio routines, and both compounds were found to be members of a family of hard phases (Hv ~ 31 GPa).

We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of the associated generalised Dirichlet-to-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems as well as in the study of their spectra., Comment: 25 pages, 1 figure

Thermal expansion of {\alpha}-rhombohedral boron ({\alpha}-B12) and two isostructural boron-rich pnictides (B12P2 and B12As2) has been studied between 298 and 1280 K by high-temperature synchrotron X-ray diffraction. For all studied phases no temperature-induced phase transitions have been observed. The observed temperature dependencies of the lattice parameters and unit cell volumes were found to be quasi-linear. Variation of the thermal expansion coefficients in the group of boron-rich pnictides (B13N2 - B12P2 - B12As2) was analyzed.

For arbitrarily small values of $\varepsilon>0,$ we formulate and analyse the Maxwell system of equations of electromagnetism on $\varepsilon$-periodic sets $S^\varepsilon\subset{\mathbb R}^3.$ Assuming that a family of Borel measures $\mu^\varepsilon,$ such that ${\rm supp}(\mu^\varepsilon)=S^\varepsilon,$ is obtained by $\varepsilon$-contraction of a fixed 1-periodic measure $\mu,$ and for right-hand sides $f^\varepsilon\in L^2({\mathbb R}^3, d\mu^\varepsilon),$ we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when $\mu$ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for $L^2$ functions, an associated Poincar\'{e}-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method., Comment: 43 pages

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media., Comment: 40 pages, 1 figure

A novel approach to critical-contrast homogenisation is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media., Comment: 37 pages, 3 figures; as accepted by Transactions of the Moscow Mathematical Society; continues 1803.09372 and supplements 1808.03961

Compressibility of boron subarsenide B12As2 has been studied by synchrotron X-ray diffraction up to 47 GPa at room temperature in a diamond anvil cell using Ne pressure transmitting medium. A fit of experimental p-V data by Vinet equation of state yielded the bulk modulus of 150(4) GPa and its first pressure derivative of 6.4(3). No pressure-induced phase transitions have been observed.

Motivated by the urgent need to attribute a rigorous mathematical meaning to the term "metamaterial", we propose a novel approach to the homogenisation of critical-contrast composites. This is based on the asymptotic analysis of the Dirichlet-to-Neumann map on the interface between different components ("stiff" and "soft") of the medium, which leads to an asymptotic approximation of eigenmodes. This allows us to see that the presence of the soft component makes the stiff one behave as a class of time-dispersive media. By an inversion of this argument, we also offer a recipe for the construction of such media with prescribed dispersive properties from periodic composites., Comment: 24 pages, 6 fugures

We analyse waves that propagate along the interface between a dielectric half-space and a half-space filled with a Lorentz material. We show that the corresponding interface condition leads to a generalisation of the classical Leontovich condition on the boundary of a dielectric half-space. We study when this condition supports propagation of (dispersive) surface waves. We derive the related dispersion relation for waves propagating along the boundary of a stratified half-space and determine the relationship between the loss parameter, frequency and wavenumber for which interfacial waves exist., Comment: 16 pages, 6 figures

Starting from the wave equation for a medium with material properties that vary periodically, we study a system of recurrence relations that describe propagation of wave packets that oscillate on the microscale (i.e. on lengths of the order of the period of the medium) and vary slowly on the macroscale (i.e. on lengths that contain a large number of periods). The resulting equations contain a version of the geometric optics and the overall energy transport description for periodic media. We illustrate the developed asymptotic theory using the example of a point pulse propagating through a periodic arrangement of two materials with highly contrasting elastic moduli., Comment: 17 pages

We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness., Comment: 39 pages

Quasi-hydrostatic compression of rhombohedral boron monosulfide (r-BS) has been studied up to 50 GPa at room temperature using diamond-anvil cells and angle-dispersive synchrotron X-ray diffraction. A fit of the experimental P-V data to the Vinet equation of state yields bulk modulus of 42.2(1.4) GPa and its first pressure derivative of 7.6(2) that are in excellent agreement with our ab initio calculations. Formation of a new high-pressure phase of boron monosulfide (hp-BS) has been observed above 35 GPa. According to ab initio evolutionary crystal structure predictions combined with Rietveld refinement of high-pressure X-ray diffraction data, the structure of hp-BS has trigonal symmetry and belongs to the space group P-3m1. As it follows from electron density of states calculations, the phase transformation is accompanied by an insulator-metal transition.

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with $\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $\mu^\varepsilon$ is the measure obtained by $\varepsilon$-scaling of $\mu.$ Our analysis includes both the case of a measure absolutely continuous with respect to the standard Lebesgue measure and the case of "singular" periodic structures (or "multistructures"), when $\mu$ is supported by lower-dimensional manifolds., Comment: 16 pages

The lattice parameters of two boron subnitrides, B13N2 and B50N2, have been measured as a function of temperature between 298 and 1273 K, and the corresponding thermal expansion coefficients have been determined. Thermal expansion of both boron subnitrides was found to be quasi-linear, and the volume thermal expansion coefficients of B50N2 (15.7(2) 10-6 K-1) and B13N2 (21.3(2) 10-6 K-1) are of the same order of magnitude as those of boron-rich compounds with structure related to {\alpha}-rhombohedral boron. For both boron subnitrides no temperature-induced phase transitions have been observed in the temperature range under study.

This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situations., Comment: 32 pages; This is the continuation of arXiv:1703.06220 (and formerly contained in v1); this version is as accepted by the journal (Operator Theory: Advances and Applications)

Using a suitable stochastic version of the compactness argument of [V. V. Zhikov, 2000. On an extension of the method of two-scale convergence and its applications. Sb. Math., 191(7--8), 973--1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled "macroscopic" and "microscopic" components of the field, giving rise to an analogue of the "Zhikov function", which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit., Comment: 24 pages

New boron subnitride B50N2 has been synthesized by crystallization from the B-BN melt at 5 GPa, and its structure has been refined using Rietveld analysis. B50N2 crystallizes in the tetragonal space group P-4n2 with unit cell parameters a = 8.8181(2) A and c = 5.0427(10) A. Quasi-hydrostatic compression of two boron subnitrides, B50N2 and B13N2, has been studied to 30 GPa at room temperature in a diamond-anvil cell using synchrotron X-ray diffraction. No pressure-induced phase transitions have been observed. A fit of experimental p-V data to the Murnaghan equation of state yielded B50N2 and B13N2 bulk moduli of 167(2) and 205(2) GPa, respectively, with fixed first bulk modulus pressure derivative of 4.0.

For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-\Delta$ in the perforated domain $\Omega\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$ $a_\varepsilon\ll\varepsilon,$ to the limit operator $-\Delta+\mu_{\iota}$ on $L^2(\Omega)$, where $\mu_\iota\in\mathbb C$ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Pro c. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem., Comment: 18 pages, 2 figures

We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier--Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces., Comment: 22 pages

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with $\delta$-type vertex conditions., Comment: 28 pages

Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\l_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely" localised to the defect, in the sense that the localisation exponent behaves as $\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing literature. As a consequence, we argue that $\l_0$ is an eigenvalue of a certain limit operator defined on the defect only. In two- and three-dimensional configurations, whose one-dimensional cross-sections are described by the setting considered, this implies the existence of propagating waves that are localised to the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.

Compressibility of boron subphosphide B12P2 has been studied under quasi-hydrostatic conditions up to 26 GPa and 2600 K using laser-heated diamond anvil cell and angle-dispersive synchrotron X-ray diffraction. 300-K data fit yields the values of bulk modulus B0 = 192(11) GPa and its first pressure derivative B0' = 5.5(12). At ambient pressure the thermal expansion is quasi-linear up to 1300 K with average volume expansion coefficient {\alpha} = 17.4(1) 10-6 K-1. The whole set of experimental p-V-T data is well described by the Anderson-Gr\"uneisen model with {\delta}T = 6.