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1. Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

   https://doi.org/10.1093/imamat/hxz019  

   Shubov, Marianna A; Kindrat, Laszlo P (October 2019, IMA Journal of Applied Mathematics)

   Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($$\alpha ,\beta ,k_1,k_2$$) linear boundary feedback law at the right end. The $$2 \times 2$$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $$\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$$. The role of the control parameters is examined and the following results have been proven: (i) when $$\beta \neq 0$$, the set of vibrational modes is asymptotically close to the vertical line on the complex $$\nu$$-plane given by the equation $$\Re \nu = \alpha + (1-k_1k_2)/\beta$$; (ii) when $$\beta = 0$$ and the parameter $$K = (1-k_1 k_2)/(k_1+k_2)$$ is such that $$\left |K\right |\neq 1$$ then the following relations are valid: $$\Re (\nu _n/n) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iii) when $$\beta =0$$, $|K| = 1$, and $$\alpha = 0$$, then the following relations are valid: $$\Re (\nu _n/n^2) = O\left (1\right )$$ and $$\Im (\nu _n/n) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iv) when $$\beta =0$$, $|K| = 1$, and $$\alpha>0$$, then the following relations are valid: $$\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$. 

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2. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai (November 2019, International Mathematics Research Notices)

   Abstract In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $$H$$ as a perturbation of the free operator $$H_0$$, where $$(H_0u)(n)= u({n+1})+u({n-1})$$. For $$H_0$$ (no perturbation), $$\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$$ and $$H_0$$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $$H_0+V$$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $$\limsup _{n\to \infty } n|V(n)|=a<\infty .$$ We obtain a lower/upper bound of $$a$$ such that $$H_0+V$$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $$\{ E_j\}_{j=1}^N$$ in $(-2,2)$ with $$0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$$, we construct the explicit potential $$V(n)=\frac{O(1)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}_{j=1}^N$$.5: Given any countable set of points $$\{ E_j\}$$ in $(-2,2)$ with $$0\notin \{ E_j\}+\{ E_j\}$$, and any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $$|V(n)|\leq \frac{h(n)}{1+|n|}$$ such that $$H=H_0+V$$ has eigenvalues $$\{ E_j\}$$. 

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3. On an inverse problem of nonlinear imaging with fractional damping

   https://doi.org/10.1090/mcom/3683  

   Kaltenbacher, Barbara; Rundell, William (January 2022, Mathematics of Computation)

   This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity κ ( x ) \kappa (x) , in what becomes a nonlinear hyperbolic equation with non-local terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from κ \kappa to the overposed data and from this basis develop and analyse Newton-type schemes for its effective recovery. 

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4. Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

   https://doi.org/10.1093/imrn/rny097  

   Liu, Wencai (May 2018, International Mathematics Research Notices)

   Abstract In this paper, we consider the eigensolutions of $$-\Delta u+ Vu=\lambda u$$, where $$\Delta $$ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato’s methods on manifold and establish the growth of the eigensolutions as r goes to infinity based on the asymptotical behaviors of $$\Delta r$$ and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $$ K_{\textrm{rad}}(r)= -1+\frac{o(1)}{r}$$. 

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5. Measured and simulated thermoelectric properties of FeAs _2−x Se _x ( x = 0.30–1.0): from marcasite to arsenopyrite structure

   https://doi.org/10.1039/d0ma00371a  

   Perez, Christopher J.; Devlin, Kasey P.; Skaggs, Callista M.; Tan, Xiaoyan; Frank, Corey E.; Badger, Jackson R.; Kang, Chang-Jong; Emge, Thomas J.; Kauzlarich, Susan M.; Taufour, Valentin; et al (August 2020, Materials Advances)

   null (Ed.)

   FeAs 2−x Se x ( x = 0.30–1.0) samples were synthesized as phase pure powders by conventional solid-state techniques and as single crystals ( x = 0.50) from chemical vapor transport. The composition of the crystals was determined to be Fe 1.025(3) As 1.55(3) Se 0.42(3) , crystallizing in the marcasite structure type, Pnnm space group. FeAs 2−x Se x (0 < x < 1) was found to undergo a marcasite-to-arsenopyrite ( P 2 1 / c space group) structural phase transition at x ∼ 0.65. The structures are similar, with the marcasite structure best described as a solid solution of As/Se, whereas the arsenopyrite has ordered anion sites. Magnetic susceptibility and thermoelectric property measurements from 300–2 K were performed on single crystals, FeAs 1.50 Se 0.50 . Paramagnetic behavior is observed from 300 to 17 K and a Seebeck coefficient of −33 μV K −1 , an electrical resistivity of 4.07 mΩ cm, and a very low κ l of 0.22 W m −1 K −1 at 300 K are observed. In order to determine the impact of the structural transition on the high-temperature thermoelectric properties, polycrystalline FeAs 2−x Se x ( x = 0.30, 0.75, 0.85, 1.0) samples were consolidated into dense pellets for measurements of thermoelectric properties. The x = 0.85 sample shows the best thermoelectric performance. The electronic structure of FeAsSe was calculated with DFT and transport properties were approximately modeled above 500 K. 

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