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Non-Hermitian matrices $H\in M_2(\mathbb{C})$ satisfying the relation $ H^{\dag}G = GH $, for invertible and singular Hermitian matrices $G$ have been studied. The matrices $H$ corresponding to invertible $G$ are known in the literature as G-pseudo Hermitian matrices. We label the matrices corresponding to the singular $G_s$ as $G_s$-pseudo Hermitian. We have proved that all $ 2\times 2$ $G$-pseudo Hermitian matrices are PT-symmetric. For a given $G$ ($G_s$), all $G$ ($G_s$)-pseudo-Hermitian $H\in M_2(\mathbb{C})$ are found to be expressed as a linear variety. It is further found that for any two Hermitian $G_i,G_j\in M_2(\mathbb{C})$ such that $G_i\neq \lambda G_j$, there always exists exactly one trace less $H\in M_2(\mathbb{C})$ (up to real scaling) which is pseudo-Hermitian with respect to both these $G$ matrices. The set of all $G$- and $G_s$- pseudo-Hermitian matrices has been divided into seven distinct ensembles of matrices and the set of all PT-symmetric matrices in $M_2(\mathbb{C})$ is partitioned into four cells, denoted by $S_1,S_2,S_3$ and $S_4$. The ensembles of trace-less G-pseudo Hermitian matrices are shown to be written as a linear combination of three basis elements from these cells. When $\mathrm{Tr}(G) = 0$, one basis element is from $S_1$ and the other two are from $S_2$. On the other hand, when $\mathrm{Tr}(G)\neq0$, one basis element is from $S_1$ and the other two are from $S_4$. The determinant of such ensembles of trace-less matrices are shown to be quadrics, which could be hyperboloid of two sheets, hyperboloid of one sheet, ellipsoid or quadric cone for invertible $G$, whereas it is two parallel planes or a plane for singular $G_s$. Finally, the set of all the matrices $G\in M_2(\mathbb{C})$, satisfying $H^{\dagger}G = GH$, given a specific $H\in M_2(\mathbb{C})$, are shown to be describable in terms of quadratic variety., Comment: This article has been submitted for peer review to the Journal of Mathematical Physics (AIP Publishing)

Carbon emissions are rising at an alarming rate, posing a significant threat to global efforts to mitigate climate change. Electric vehicles have emerged as a promising solution, but their reliance on lithium-ion batteries introduces the critical challenge of battery degradation. Accurate prediction and forecasting of battery degradation over both short and long time spans are essential for optimizing performance, extending battery life, and ensuring effective long-term energy management. This directly influences the reliability, safety, and sustainability of EVs, supporting their widespread adoption and aligning with key UN SDGs. In this paper, we present a novel approach to the prediction and long-term forecasting of battery degradation using Scientific Machine Learning framework which integrates domain knowledge with neural networks, offering more interpretable and scientifically grounded solutions for both predicting short-term battery health and forecasting degradation over extended periods. This hybrid approach captures both known and unknown degradation dynamics, improving predictive accuracy while reducing data requirements. We incorporate ground-truth data to inform our models, ensuring that both the predictions and forecasts reflect practical conditions. The model achieved MSE of 9.90 with the UDE and 11.55 with the NeuralODE, in experimental data, a loss of 1.6986 with the UDE, and a MSE of 2.49 in the NeuralODE, demonstrating the enhanced precision of our approach. This integration of data-driven insights with SciML's strengths in interpretability and scalability allows for robust battery management. By enhancing battery longevity and minimizing waste, our approach contributes to the sustainability of energy systems and accelerates the global transition toward cleaner, more responsible energy solutions, aligning with the UN's SDG agenda.

In this paper, we introduce fastHDMI, a Python package designed for efficient variable screening in high-dimensional datasets, particularly neuroimaging data. This work pioneers the application of three mutual information estimation methods for neuroimaging variable selection, a novel approach implemented via fastHDMI. These advancements enhance our ability to analyze the complex structures of neuroimaging datasets, providing improved tools for variable selection in high-dimensional spaces. Using the preprocessed ABIDE dataset, we evaluate the performance of these methods through extensive simulations. The tests cover a range of conditions, including linear and nonlinear associations, as well as continuous and binary outcomes. Our results highlight the superiority of the FFTKDE-based mutual information estimation for feature screening in continuous nonlinear outcomes, while binning-based methods outperform others for binary outcomes with nonlinear probability preimages. For linear simulations, both Pearson correlation and FFTKDE-based methods show comparable performance for continuous outcomes, while Pearson excels in binary outcomes with linear probability preimages. A comprehensive case study using the ABIDE dataset further demonstrates fastHDMI's practical utility, showcasing the predictive power of models built from variables selected using our screening techniques. This research affirms the computational efficiency and methodological strength of fastHDMI, significantly enriching the toolkit available for neuroimaging analysis., Comment: 31 pages, 5 figures

Ly$\alpha$ emission is key to understanding the process of cosmic reionisation. JWST is finally enabling us to measure Ly$\alpha$ emission deep into the epoch of reionisation for an increasing number of galaxies. However, discrepancies between measurements of Ly$\alpha$ equivalent widths (EW$_0$) of Ly$\alpha$ emitters (LAEs) have been noted between JWST and ground-based facilities. We employ SPICE, a suite of radiation-hydrodynamical simulations featuring different stellar feedback models, and investigate the impact of radiative transfer effects and observational systematics on the measured Ly$\alpha$ EW$_0$. We perform radiative transfer of Ly$\alpha$ and UV photons for SPICE galaxies to mimic slit spectroscopy for ground-based slits and JWST-MSA pseudo-slits. Spatial Ly$\alpha$-UV offsets exist independently of feedback model, and are common ($>70$% galaxies) with median values of $\approx0.07-0.11''$ and these predictions are consistent with the observed spatial offsets. Spatial Ly$\alpha$-UV offsets are a major cause of loss of flux for JWST-MSA observations, with median pseudo-slit losses of $\approx65$%, and $\approx30$% cases suffering from $>95$% pseudo-slit losses. Even in the absence of such spatial offsets, the presence of extended emission can cause median pseudo-slit losses of $40$%, with $4$% cases suffering from $>95$% pseudo-slit losses. Complex galaxy morphologies or misplaced JWST-MSA pseudo-slit can lead to under-estimated UV continuum, resulting in spuriously high estimates of EW$_0$ from JWST in $6-8$% of galaxies. We compare the predictions from {\tt SPICE} to 25 galaxies with Ly$\alpha$ emission observations from both the ground and JWST. The EW$_0^{\tt JWST}$ and the EW$_0^{\tt Ground}$ exhibit scatter in line with predictions, indicating that both physical and systematic effects are likely at play.

Tunneling half lives are obtained in a minimalistic deformation picture of nuclear decays. As widely documented in other deformation models, one finds that the effective mass of the nucleus changes with the deformation parameter. However, contrary to the approach used in literature, a position-dependant mass potentially makes using WKB tunneling probabilities unreliable for estimating nuclear lifetimes. We instead use a new approach, a combination of the Transmission Matrix and WKB methods, to estimate tunneling probabilities. Because of the simplistic nature of the model, the calculated lifetimes are not accurate, however, the relative trends in the lifetimes of isotopes of individual nuclei are found to be consistent. Using this, we develop an empirical scaling to obtain the actual half-lives, and find the primary scaling parameter to have remarkably consistent values for all nuclei considered. The new tunneling method proposed here, which produces very different probabilities as compared to the usual WKB approach, is another key result of this work, and can be utilized for arbitrary potentials and mass variations.

In any extension of General Relativity (GR), extra fundamental degrees of freedom couple to gravity. Besides deforming GR forecasts in a theory-dependent way, this coupling generically introduces extra modes in the gravitational-wave signal. We propose a novel theory-agnostic test of gravity to search for these nongravitational modes in black hole merger ringdown signals. To leading order in the GR deviations, their frequencies and damping times match those of a test scalar or vector field in a Kerr background, with only amplitudes and phases as free parameters. This test will be highly valuable for future detectors, which will achieve signal-to-noise ratios higher than 100 (and as high as 1000 for space-based detectors such as LISA). Such sensitivity will allow measurement of these modes with amplitude ratios as low as 0.05 for ground-based detectors (and as low as 0.008 for LISA), relative to the fundamental mode, enabling stringent agnostic constraints or detection of scalar/vector modes. By applying this test to GW150914, GW190521, and GW200129, we find that the current evidence for an extra mode is comparable to that for the first gravitational overtone, but its inclusion modifies the inferred remnant spin., Comment: 10 pages, 9 figures. v2: minor modifications, new explicit example, and new results on precessing binaries. Submitted version

The remnant black hole from a binary coalescence emits ringdown gravitational waves characterized by quasi-normal modes, which depend solely on the remnant's mass and spin. In contrast, the ringdown amplitudes and phases are determined by the properties of the merging progenitors. Accurately modeling these amplitudes and phases reduces systematic biases in parameter estimation and enables the development and performance of rigorous tests of general relativity. We present a state-of-the-art, data-driven surrogate model for ringdown amplitudes and phases, leveraging Gaussian Process Regression trained against SXS numerical-relativity simulations. Focusing on non-precessing, quasi-circular binary black holes, our model offers the most comprehensive fit that includes 16 emission modes, incorporating overtones and quadratic contributions. Our surrogate model achieves reconstruction errors that are approximately two orders of magnitude smaller than the typical measurement errors of current gravitational-wave interferometers. An additional benefit of our approach is its flexibility, which allows for future extensions to include features such as eccentricity and precession, broadening the scope of its applicability to more generic astrophysical scenarios. Finally, we are releasing our model in a ready-to-use package called postmerger., Comment: 12 pages, 7 figures; code usage examples are available at https://github.com/cpacilio/postmerger

This paper focuses on the challenging task of bottom-up restoration in a complete blackout system using Grid-forming (GFM) converters. Challenges arise due to the limited current capability of power converters, resulting in distinct dynamic responses and fault current characteristics compared to synchronous generators. Additionally, GFM control needs to address the presence of unbalanced conditions commonly found in distribution systems. To address these challenges, this paper explores the black start capability of GFM converters with a generalized three-phase GFM droop control. This approach integrates GFM controls individually for each phase, incorporating phase-balancing feedback and enabling current limiting for each phase during unbalanced faults or overloading. The introduction of a phase-balancing gain provides flexibility to trade-off between voltage and power imbalances. The study further investigates bottom-up black start operations using GFM converters, incorporating advanced load relays into breakers for gradual load energization without central coordination. The effectiveness of bottom-up black start operations with GFM converters, utilizing the generalized three-phase GFM droop, is evaluated through electromagnetic transient (EMT) simulations in MATLAB/Simulink. The results confirm the performance and effectiveness of this approach in achieving successful black start operations under unbalanced conditions.

Let $(\tau, V_{\tau})$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $\Gamma$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $\tau$ and multiplicities of irreducible $\tau^\vee$-spherical spectra in $L^2(\Gamma \backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $\tau$-isospectrality. Along with this we prove that the almost equality of $\tau$-spherical spectra of two lattices assures the equality of their $\tau$-spherical spectra., Comment: 18 pages, minor revisions

This article is inspired from the work of M Krithika and P Vanchinathan on Cluster Magnification and the work of Alexander Perlis on Cluster Size. We establish the existence of polynomials for given degree and cluster size over number fields which generalises a result of Perlis. We state the Strong cluster magnification problem and establish an equivalent criterion for that. We also discuss the notion of weak cluster magnification and prove some properties. We provide an important example answering a question about Cluster Towers. We introduce the concept of Root capacity and prove some of its properties. We also introduce the concept of unique descending and ascending chains for extensions and establish some properties and explicitly compute some interesting examples. Finally we establish results about all these phenomena under a particular type of base change. The article concludes with results about strong cluster magnification and unique chains and some properties of the ascending index for a field extension., Comment: 34 pages. Changes in v3 : Significant changes in old sections of v2. New section on ascending index added to v2. Changes in v2 : Changed the title (previous title was "Inverse Cluster Magnifi..."). Significant changes in old sections of v1. New sections on root capacity, unique chains and base change added to v1. Submitted to Journal of Number Theory

Gravitational waves emitted by a ringing black hole allow us to perform precision tests of general relativity in the strong field regime. With improvements to our current gravitational wave detectors and upcoming next-generation detectors, developing likelihood-free parameter inference infrastructure is critical as we will face complications like nonstandard noise properties, partial data and incomplete signal modeling that may not allow for an analytically tractable likelihood function. In this work, we present a proof-of-concept strategy to perform likelihood-free Bayesian inference on ringdown gravitational waves using simulation based inference. Specifically, our method is based on truncated sequential neural posterior estimation, which trains a neural density estimator of the posterior for a specific observed data segment. We setup the ringdown parameter estimation directly in the time domain. We show that the parameter estimation results obtained using our trained networks are in agreement with well-established Markov-chain methods for simulated injections as well as analysis on real detector data corresponding to GW150914. Additionally, to assess our approach's internal consistency, we show that the density estimators pass a Bayesian coverage test., Comment: 12 pages, 7 figures; v2: minor changes, matches version to appear on Phys.Rev.D

In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois representations., Comment: 21 pages

We present an extension of the RSVD-$\Delta t$ algorithm initially developed for resolvent analysis of statistically stationary flows to handle harmonic resolvent analysis of time-periodic flows. The harmonic resolvent operator, as proposed by \citet{Padovanetal20}, characterizes the linearized dynamics of time-periodic flows in the frequency domain, and its singular value decomposition reveals forcing and response modes with optimal energetic gain. However, computing harmonic resolvent modes poses challenges due to $(i)$ the coupling of all $N_{\omega}$ retained frequencies into a single harmonic resolvent operator and $(ii)$ the singularity or near-singularity of the operator, making harmonic resolvent analysis considerably more computationally expensive than a standard resolvent analysis. To overcome these challenges, the RSVD-$\Delta t$ algorithm leverages time stepping of the underlying time-periodic linearized Navier-Stokes operator, which is $N_{\omega}$ times smaller than the harmonic resolvent operator, to compute the action of the harmonic resolvent operator. We develop strategies to minimize the algorithm's CPU and memory consumption, and our results demonstrate that these costs scale linearly with the problem dimension. We validate the RSVD-$\Delta t$ algorithm by computing modes for a periodically varying Ginzburg-Landau equation and demonstrate its performance using the flow over an airfoil.

A geometrical analysis of the stability of nuclei against deformations is presented. In particular, we use Catastrophe Theory to illustrate discontinuous changes in the behavior of nuclei with respect to deformations as one moves in the N - Z space. We construct a minimalistic deformation model using the microscopic-macroscopic approach. A third-order phase transition is found in the liquid-drop model, which translates to a complete loss of stability (using the Fold catastrophe) when shell effects are included. The analysis is found to explain the instability of known fissile nuclei and also justify known decay chains of heavy nuclei., Comment: 16 pages, 3 figures, 2 tables

The applications of terahertz metamaterials are being actively explored in recent times for applications in high-speed communication devices, miniature photonic circuits, and bio-chemical devices because of their wide advantages. The toroidal resonance, a new type of metasurface resonance, has been examined with great interest to utilize its properties in terahertz metasurface applications. The study reports a proof of concept design of a toroidal metasurface that experimentally demonstrates logic gate operations in the terahertz frequency regime by passive tuning of the split ring resonators compromising the meta-atom design. The amplitude modulation is utilized as a method of determining the Boolean logic output of the system. The proposed metasurface could be further optimized for high amplitude modulations, as well as for active logic gate operations using tunable materials including graphene and ITO. The proposed metasurface consists of three split-ring resonators, and the near-field coupling between the adjacent resonators dictates the logic gate operations. The toroidal excitation in the metasurface is determined by a multipole analysis of the scattered powers of terahertz radiation. The proposed metasurfaces experimentally define AND Boolean logic gate at 0.89 terahertz, and OR Boolean logic gate at 0.97 terahertz. Numerical simulations support the experimentally obtained results. Additionally, we numerically report the excitation of NAND gate at 0.87 THz. Such toroidal logic-gate metasurfaces in the terahertz region could find applications in digitized terahertz circuits for photonic device applications.

We present SPICE, a new suite of RHD cosmological simulations targeting the epoch of reionisation. The goal of these simulations is to systematically probe a variety of stellar feedback models, including "bursty" and "smooth" forms of supernova energy injection, as well as poorly-explored scenarios such as hypernova explosions and radiation pressure. Subtle differences in the behaviour of supernova feedback drive profound differences in reionisation histories, with burstier forms of feedback causing earlier reionisation. We also find that some global galaxy properties, such as the dust-attenuated luminosity functions and star formation main sequence, remain degenerate between models. Stellar feedback and its strength determine the morphological mix of galaxies emerging by z = 5 and that the reionisation history is inextricably connected to intrinsic properties such as galaxy kinematics and morphology. While star-forming, massive disks are prevalent if supernova feedback is "smooth", "bursty" feedback preferentially generates dispersion-dominated systems. Different modes of feedback produce different strengths of outflows, altering the ISM/CGM in different ways, and in turn strongly affecting the escape of LyC photons. We establish a correlation between galaxy morphology and LyC escape fraction, revealing that dispersion-dominated systems have escape fractions 10-50 times higher than their rotation-dominated counterparts at all redshifts. Dispersion-dominated systems should thus preferentially generate large HII regions as compared to their rotation-dominated counterparts. Since dispersion-dominated systems are more prevalent if stellar feedback is more explosive, reionisation occurs earlier in our simulation with burstier feedback. Statistical samples of post-reionisation galaxy morphologies probed with JWST, ALMA and MUSE can constrain stellar feedback and models of cosmic reionisation., Comment: arXiv admin note: text overlap with arXiv:1801.07259 by other authors

Combining multiple gravitational-wave observations allows for stringent tests of general relativity, targeting effects that would otherwise be undetectable using single-event analyses. We highlight how the finite size of the observed catalog induces a significant source of variance. If not appropriately accounted for, general relativity can be excluded with arbitrarily large credibility even if it is the underlying theory of gravity. This effect is generic and holds for arbitrarily large catalogs. Moreover, we show that it cannot be suppressed by selecting "golden" observations with large signal-to-noise ratios. We present a mitigation strategy based on bootstrapping (i.e. resampling with repetition) that allows assigning uncertainties to one's credibility on the targeted test. We demonstrate our findings using both toy models and real gravitational-wave data. In particular, we quantify the impact of the catalog variance on the ringdown properties of black holes using the latest LIGO/Virgo catalog., Comment: 6 pages, 2 figures; v2: matches published version

We prove an analogue of the strong multiplicity one theorem in the context of finite covolume lattices in the group $G ={ \rm SO}(2,1)^\circ$. We show that the multiplicity measures of two lattices are same if they agree outside a finite measure subset of $\widehat G$., Comment: 15 Pages

In this work, we investigate grid-forming (GFM) control for dc/ac voltage source converters (VSC) under unbalanced system conditions and unbalanced faults. To fully leverage the degrees of freedom of VSCs, we introduce the concept of generalized three-phase GFM control that combines individual GFM controls for every phase with a phase balancing feedback. The proposed control allows trading off voltage and power unbalance under unbalanced conditions, enables current limiting for each phase during unbalanced faults, and reduces to positive sequence GFM droop control in balanced systems. High-fidelity simulations are used to illustrate the properties of the control.

A geometric perspective of the Higgs Mechanism is presented. Using Thom's Catastrophe Theory, we study the emergence of the Higgs Mechanism as a discontinuous feature in a general family of Lagrangians obtained by varying its parameters. We show that the Lagrangian that exhibits the Higgs Mechanism arises as a first-order phase transition in this general family. We find that the Higgs Mechanism (as well as Spontaneous Symmetry Breaking) need not occur for a different choice of parameters of the Lagrangian, and further analysis of these unconventional parameter choices may yield interesting implications for beyond standard model physics., Comment: 5 figures

Following the upgrades to Advanced LIGO (aLIGO), measurements were made of the detector suspensions' frequency response characteristics. While most resonant frequencies could be identified with simple mechanical models, such as the fiber vibration modes, some were unexplained. Using a finite element model of the quadruple pendulum suspension, we search for and identify lines from unknown sources. The present work focuses on two resonant lines observed in the Upper Intermediate Mass as examples of this technique. Our simulations suggest a common source for these lines, which could be accounted for in a redesign. By modeling these response frequencies, we can examine the motion of individual components, and suggest methods to reduce their amplitude, alter their frequency, or eliminate them in future gravitational wave detector designs.