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Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$. The null controllability properties of the equation $$u_{tt}+\Delta^2 u+\rho (\Delta)^\alpha u_t=F(x,t)$$ are studied. Let $T>0$, and assume initial conditions $(u^0,u^1)\in Dom(\Delta)\times L^2(0,\pi)$. We first prove finite dimensional null control results: suppose $F(x,t)=f^1(t)h^1(x)+f^2(t)h^2(x)$ with $h^1,h^2$ given functions. For $\alpha \in [0,3/2)$, we prove that there exist $h^1,h^2\in L^2(0,\pi)$ such that for any $(u^0,u^1)$, there exist $L^2$ null controls $(f^1,f^2).$ For $\alpha< 1$ and $\rho <2$, we prove null controllability with $f^2=0$ and $h^1$ belonging to a large class of functions. For $\alpha\in [3/2,2)$, we prove spectral and null controllability both generally fail, but two dimensional weak controllability holds. Our second set of results pertains to $F(x,t)=\chi_\Omega(x)f(x,t)$, with $\Omega$ any open subset of $(0,\pi)$. For any $\alpha \in [0,3/2),$ we prove there exists a null control $f\in L^2(\Omega\times(0,T))$ To prove our main results, we use the Fourier method to rewrite the control problems as moment problems. These are then solved by constructing biorthogonal sets to the associated exponential families. These constructions seem to be non-standard and may be of independent interest.

We present high field magnetotransport studies of Ge1-x-y(SnxMny)Te bulk multiferroics with diamagnetic Sn and paramagnetic Mn concentration x = 0.38 to 0.79 and y = 0.02 to 0.086, respectively. The zero field resistivity, {\rho}(T) takes significant contribution from defects below T = 20 K however, a mixed scattering contribution from dynamic disorder and unusual sources is estimated from T = 20 K to 300 K. The carrier mobility shows anomalous temperature dependence from T0.2 to T0.5 which hints towards possible presence of polaronic effects resulting from coupling of holes with phonons. This anomalous behavior cannot be understood in terms of pure phononic scattering mechanism at high temperature. From point of view of high field results, the transverse component of magnetoresistivity manifests anomalous Hall effect originating from extrinsic scattering sources, particularly the side jump mechanism reveals a larger contribution. We also find that the correlation between the transverse and longitudinal conductivities follow the universal scaling law {\sigma}xy ~ {\sigma}xxn where n = 1.6 in the low conductivity limit. The values n = 1.5 to 1.8 obtained for the present GSMT alloys justify the bad metal hopping regime since the results fall in the low conductivity ferromagnetic family with {\sigma}xx ~ 104 ohmcm-1. The interpretation of the n = 1.6 scaling in the low conductivity regime is thus far not fully understood. However, the anomalous Hall resistivity scaling with modified relation by Tian et al is indicative of the dominant side jump scattering along with noticeable role of skew scattering., Comment: 13 pages, 9 figures, 2 tables

A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step., Comment: arXiv admin note: text overlap with arXiv:2302.05970

We report the structural and magnetic results of polar {\alpha}-GeTe doped with Sn and Mn from x = 0.185 to 0.841 and y = 0.02 to 0.086, respectively. The magnetic results of Ge1-x-y(SnxMny)Te (GSMT) crystals identify Mn-clustering effect with scaling parameter, R = 0.033 for x ~ 0.2 and y = 0.06. The excessive Sn ions are assumed to drive the inception of short range ferromagnetic clusters. For the crystals revealing glassy magnetic behavior, the irreversibility temperature, Tirr shifts at high dc magnetic field which is well described by de Almeida-Thouless equation, {\delta}Tirr ~ H{\Phi}/2 yielding {\Phi} values of 1.55 and 1.7 that validates the formation of Mn clusters. The spin relaxation time, {\tau}0 ~ 10-9 s, activation energy, Ea/kB ~ 5 TF where TF is freeing temperature and Vogel-Fulcher temperature, T0 ~ TF also signify intracluster interactions. Also, the Mn-hole magnetic exchange constant, Jpd drops from 0.24 eV for the samples with smaller Sn content of x ~ 0.2 to 0.16 eV for Sn rich alloy with x ~ 0.8. Consequently, we present a magnetic phase diagram for GSMT bulk crystals as a function of Mn(y) ions., Comment: This preprint has 18 pages, 8 main figures, 2 tables and 7 supporting figures

The inverse problem of recovery of a potential on a quantum tree graph from Weyl's matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two-spectra inverse problems on separate edges and reduce them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests., Comment: arXiv admin note: substantial text overlap with arXiv:2210.15536

In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff-Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and moment method approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph.

In this paper, we consider the inverse dynamic problem for the Dirac system on finite metric tree graphs. Our main goal is to recover the topology (connectivity) of a tree, lengths of edges, and a matrix potential function on each edge. We use the dynamic response operator as our inverse data and apply the Leaf peeling method. In addition, we present a new dynamic algorithm to solve the forward problem for the Dirac system on general finite metric graphs.

The problem of recovery of a potential on a quantum star graph from Weyl's matrix given at a finite number of points is considered. A method for its approximate solution is proposed. It consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge with its posterior solution. The overall approach is based on Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations, and, in fact, the solution of the inverse problem on the quantum graph reduces to dealing with the NSBF coefficients. The NSBF representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the NSBF representation alone is sufficient for the recovery of the potential. The knowledge of the Weyl matrix at a set of points allows one to calculate a number of the NSBF coefficients at the end point of each edge, which leads to approximation of characteristic functions of two Sturm-Liouville problems and allows one to compute the Dirichlet-Dirichlet and Neumann-Dirichlet spectra on each edge. In turn, for solving this two-spectra inverse Sturm-Liouville problem a system of linear algebraic equations is derived for computing the first NSBF coefficient and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests., Comment: arXiv admin note: substantial text overlap with arXiv:2210.12500

A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm-Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

The question of what conditions should be set at the nodes of a discrete graph for the wave equation with discrete time is investigated. The variational method for the derivation of these conditions is used. A parallel with the continuous case is also drawn. As an example the problem of shape controllability from the boundary is studied., Comment: 15 pages, 2 figures

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.

B.\,Ya.\,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's result to a more general class of entire functions $F(z)$ with zeros in a strip $0

The paper presents a workflow for fast pore-scale simulation of single-phase flow in tight reservoirs typically characterized by low, multiscale porosity. Multiscale porosity implies that the computational domain contains porous voxels (unresolved porosity) in addition to pure fluid voxels. In this case, the Stokes-Brinkman equations govern the flow, with the Darcy term needed to account for the flow in the porous voxels. As the central part of our workflow, robust and efficient solvers for Stokes and Stokes-Brinkman equations are presented. The solvers are customized for low-porosity binary and multiclass images, respectively. Another essential component of the workflow is a preprocessing module for classifying images with respect to the connectivity of the multiscale pore space. Particularly, an approximation of the Stokes-Brinkman problem, namely, the Darcy problem, is investigated for the images that do not have pure fluid percolation paths. Thorough computational experiments demonstrate efficiency and robustness of the workflow for simulations on images from tight reservoirs. Raw files describing the used CT images are provided as supplementary materials to enable other researchers to use them.

MnBi$_{2}$Te$_{4}/$(Bi$_{2}$Te$_{3}$)$_{n}$ materials system has recently generated strong interest as a natural platform for realization of the quantum anomalous Hall (QAH) state. The system is magnetically much better ordered than substitutionally doped materials, however, the detrimental effects of certain disorders are becoming increasingly acknowledged. Here, from compiling structural, compositional, and magnetic metrics of disorder in ferromagnetic MnBi$_{2}$Te$_{4}/$(Bi$_{2}$Te$_{3}$)$_{n}$ it is found that migration of Mn between MnBi$_{2}$T$e_{4}$ septuple layers (SLs) and otherwise non-magnetic Bi$_{2}$Te$_{3}$ quintuple layers (QLs) has systemic consequences - it induces ferromagnetic coupling of Mn-depleted SLs with Mn-doped QLs, seen in ferromagnetic resonance as an acoustic and optical resonance mode of the two coupled spin subsystems. Even for a large SL separation (n $\gtrsim$ 4 QLs) the structure cannot be considered as a stack of uncoupled two-dimensional layers. Angle-resolved photoemission spectroscopy and density functional theory studies show that Mn disorder within an SL causes delocalization of electron wavefunctions and a change of the surface bandstructure as compared to the ideal MnBi$_{2}$Te$_{4}/$(Bi$_{2}$Te$_{3}$)$_{n}$. These findings highlight the critical importance of inter- and intra-SL disorder towards achieving new QAH platforms as well as exploring novel axion physics in intrinsic topological magnets., Comment: 16 pages, 5 figures, corrected typos, corrected references to figures for section 5

This paper is devoted to prove the exact controllability of a system of N one-dimensional coupled wave equations when the control is exerted on a part of the boundary by means of one control. We consider the case where the coupling matrix A has distinct eigenvalues. We give a Kalman condition (necessary and sufficient) and give a description, non-optimal in general, of the attainable set.

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for a $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results.

We study inverse boundary problems for a one dimensional linear integro-differential equation of the Gurtin--Pipkin type with the Dirichlet-to-Neumann map as the inverse data. Under natural conditions on the kernel of the integral operator, we give the explicit formula for the solution of the problem with the observation on the semiaxis $t>0.$ For the observation on finite time interval, we prove the uniqueness result, which is similar to the local Borg--Marchenko theorem for the Schr\"odinger equation., Comment: The improved version. Submitted to Inverse Problems and Imaging 11 pages, 17 references