Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized $H^{\alpha/2}$-norm of the approximated solution, where $\alpha \in [1,2)$. We demonstrate that the scheme converges strongly in $L^2(0,T;L^2_{loc}(\mathbb{R}))$ to a weak solution of the fractional KdV equation provided the initial data in $L^2(\mathbb{R})$. Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations

Nematic Fluctuations in Iron-Oxychalcogenide Mott Insulators

Nematic fluctuations occur in a wide range of physical systems from liquid crystals to biological molecules to solids such as exotic magnets, cuprates and iron-based high-$T_c$ superconductors. Nematic fluctuations are thought to be closely linked to the formation of Cooper-pairs in iron-based superconductors. It is unclear whether the anisotropy inherent in this nematicity arises from electronic spin or orbital degrees of freedom. We have studied the iron-based Mott insulators La$_{2}$O$_{2}$Fe$_{2}$O$M$$_{2}$ $M$ = (S, Se) which are structurally similar to the iron pnictide superconductors. They are also in close electronic phase diagram proximity to the iron pnictides. Nuclear magnetic resonance (NMR) revealed a critical slowing down of nematic fluctuations as observed by the spin-lattice relaxation rate ($1/T_1$). This is complemented by the observation of a change of electrical field gradient over a similar temperature range using M\"ossbauer spectroscopy. The neutron pair distribution function technique applied to the nuclear structure reveals the presence of local nematic $C_2$ fluctuations over a wide temperature range while neutron diffraction indicates that global $C_{4}$ symmetry is preserved. Theoretical modeling of a geometrically frustrated spin-$1$ Heisenberg model with biquadratic and single-ion anisotropic terms provides the interpretation of magnetic fluctuations in terms of hidden quadrupolar spin fluctuations. Nematicity is closely linked to geometrically frustrated magnetism, which emerges from orbital selectivity. The results highlight orbital order and spin fluctuations in the emergence of nematicity in Fe-based oxychalcogenides. The detection of nematic fluctuation within these Mott insulator expands the group of iron-based materials that show short-range symmetry-breaking

Nematic fluctuations in iron-oxychalcogenide Mott insulators

Nematic fluctuations occur in a wide range physical systems from biological molecules to cuprates and iron pnictide high-Tc superconductors. It is unclear whether nematicity in pnictides arises from electronic spin or orbital degrees of freedom. We studied the iron-based Mott insulators La2O2Fe2OM2M = (S, Se), which are structurally similar to pnictides. Nuclear magnetic resonance revealed a critical slowing down of nematic fluctuations and complementary Mössbauerr spectroscopy data showed a change of electrical field gradient. The neutron pair distribution function technique detected local C2 fluctuations while neutron diffraction indicates that global C4 symmetry is preserved. A geometrically frustrated Heisenberg model with biquadratic and single-ion anisotropic terms provides the interpretation of the low temperature magnetic fluctuations. The nematicity is not due to spontaneous orbital order, instead it is linked to geometrically frustrated magnetism based on orbital selectivity. This study highlights the interplay between orbital order and spin fluctuations in nematicity

Noninvasive Non-Contact SpO₂ Monitoring Using an Integrated Polarization-Sensing CMOS Imaging Sensor

Background:In the diagnosis and primary health care of an individual, estimation of the pulse rate and blood oxygen saturation (SpO2) is critical. The pulse rate and SpO2 are determined by methods including photoplethysmography (iPPG), light spectroscopy, and pulse oximetry. These devices need to be compact, non-contact, and noninvasive for real-time health monitoring. Reflection-based iPPG is becoming popular as it allows non-contact estimation of the heart rate and SpO2. Most iPPG methods capture temporal data and form complex computations, and thus real-time measurements and spatial visualization are difficult. Method:In this research work, reflective mode polarized imaging-based iPPG is proposed. For polarization imaging, a custom image sensor with wire grid polarizers on each pixel is designed. Each pixel has a wire grid of varying transmission axes, allowing phase detection of the incoming light. The phase information of the backscattered light from the fingertips of 12 healthy volunteers was recorded in both the resting as well as the excited states. These data were then processed using MATLAB 2021b software. Results: The phase information provides quantitative information on the reflection from the superficial and deep layers of skin. The ratio of deep to superficial layer backscattered phase information is shown to be directly correlated and linearly increasing with an increase in the SpO2 and heart rate. Conclusions: The phase-based measurements help to monitor the changes in the resting and excited state heart rate and SpO2 in real time. Furthermore, the use of the ratio of phase information helps to make the measurements independent of the individual skin traits and thus increases the accuracy of the measurements. The proposed iPPG works in ambient light, relaxing the instrumentation requirement and helping the system to be compact and portable