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Generalized Locally Toeplitz (GLT) matrix sequences arise from large linear systems that approximate Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), and Integro-Differential Equations (IDEs). GLT sequences of matrices have been developed to study the spectral/singular value behaviour of the numerical approximations to various PDEs, Fades and IDEs. These approximations can be achieved using any discretization method on appropriate grids through local techniques such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, and Discontinuous Galerkin methods. Spectral and singular value symbols are essential for analyzing the eigenvalue and singular value distributions of matrix sequences in the Weyl sense. In this article, we provide a comprehensive overview of the operator-theoretic aspect of GLT sequences. The theory of GLT sequences, along with findings on the asymptotic spectral distribution of perturbed matrix sequences, is a highly effective and successful method for calculating the spectral symbol f. Therefore, developing an automatic procedure to compute the spectral symbols of these matrix sequences would be advantageous, a task that Ahmed Ratnani, N S Sarathkumar, S. Serra-Capizzano have partially undertaken. As an application of the theory developed here, we propose an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming they form a GLT sequence that meets mild conditions.

We investigate the effect of structural deformation on the magnetic properties of orthorhombic CePdAl$_3$ in relation to its tetragonal polymorph. Utilizing x-ray and neutron diffraction we establish that the crystal structure has the $Cmcm$ space group symmetry and exhibits pseudo-tetragonal twinning. According to density-functional calculations the tetragonal-orthorhombic deformation mechanism has its grounds in relatively small free enthalpy difference between the polymorphs, allowing either phase to be quenched and fully accounts for the twinned microstructure of the orthorhombic phase. Neutron diffraction measurements show that orthorhombic CePdAl$_3$ establishes long-range magnetic order below $T_\mathrm{N}$=5.29 (5) K characterized by a collinear, antiferromagnetic arrangement of magnetic moments. Magnetic anisotropies of orthorhombic CePdAl$_3$ arise from strong spin-orbit coupling as evidenced by the crystal-field splitting of the $4f$ multiplet, fully characterised with neutron spectroscopy. We discuss the potential mechanism of frustration posed by antiferromagnetic interactions between nearest neighbours in the tetragonal phase, which hinders the formation of long-range magnetic order in tetragonal CePdAl$_3$. We propose that orthorhombic deformation releases the frustration and allows for long-range magnetic order., Comment: Finalized paper from the splitting of arxiv.org/abs/2106.08194v1

Natural starch-based blood-mimicking fluid (BMF) has been used as an alternative to commercially available BMFs for in-vitro Doppler investigations in low-resource settings. Most reported works in the literature have used corn starch-based BMF. Evaluation of other natural starches for potential BMF and their characterization have relatively been unexplored in the literature. To this end, this work investigates the echogenicity of corn-, potato-, tapioca-, and wheat starch-based BMFs prepared using a liquid base of pure water-glycerol mixture with three different starch concentrations (1%, 3%, and 5%). The experiments were performed by manually pumping the BMFs to a PolyVinyl alcohol (PVA) based flow phantom using a syringe and raw datasets were acquired using a Verasonics Vantage 128 Research Ultrasound System. Echogenicity was measured as the mean pixel intensity in a selected region of interest (ROI) in the beamformed image. Among the four natural starch-based BMFs, potato starch-based BMF showed the highest echogenicity and contrast with almost 13%, 14%, and 10% higher pixel intensities (dB) than that of the least echoic BMF with 1%, 3%, and 5% starch concentrations respectively. Moreover, the echogenicity of corn, tapioca, and wheat starch-based BMF was observed to be similar, and the results suggest that these BMFs with higher starch concentrations shall be employed for in-vitro contrast-enhanced ultrasound imaging., Comment: 5 pages, 4 figures. Accepted in the IEEE South Asian Ultrasonics Symposium 2024 (IEEE SAUS 2024)

This article investigates the convergence properties of Grunwald-type operators defined using the Lagrange interpolation operator. In $C[0, \pi]$, we establish sufficient conditions under which alternate grid point selection yields a convergence result. Employing a quantitative approach grounded in the modulus of continuity, we derive a theorem that provides insightful estimates for the convergence of Grunwald operator, accompanied by illustrative numerical examples. Addressing challenges related to continuity, we extend the operator to $L^1(\mathbb{R})$ and prove similar convergence results. Notably, the new operator demonstrates a uniform order of convergence on a dense subspace of $L^1(\mathbb{R})$. Moreover, it has a better convergence rate than Grunwald operator. We establish the non positivity of the Lagrange interpolation operator and, consequently, the constructed operators. Significant emphasis is placed on D.E. Wulbert theorem ($1968$), which is a Korovkin-type theorem that can be applied to non-positive operators. We extend this theorem to a broader class of operators, enabling its effective application to the constructed operators. This article, overall, highlights the broad applicability of Lagrange interpolation across diverse disciplines.