The particle nature of dark matter (DM) remains one of the most significant enigmas in modern cosmology. Axion-like particles (ALPs), as well-motivated candidates for cold dark matter, can undergo radiative decay into photon pairs, a process that is significantly enhanced in the presence of ambient radiation fields. In this work, we propose a novel probe of $μ{\rm eV}$-scale ALP DM by cross-correlating radio intensity mapping (IM) with the large-scale galaxy distribution from the 2MASS Redshift Survey (2MRS) in the local universe ($z\leq 0.1$). We develop a comprehensive theoretical framework that incorporates stimulated decay effects driven by both the Cosmic Microwave Background (CMB) and a bottom-up modeled extragalactic radio background (ERB). By forecasting the sensitivity of the Square Kilometre Array (SKA) Phase 2, we demonstrate that this cross-correlation technique provides a promising and complementary approach to searching for ALP DM signals. This study establishes a new proof-of-concept for utilizing next-generation radio telescopes to probe ALP dark matter on cosmic scales.

Coronal hole boundaries are the interfaces between closed and open magnetic field regions in the solar atmosphere. Many fundamental processes take place at these regions, including magnetic reconnection that is responsible for solar wind release and restructuring of the solar magnetic field. In this paper, we present a case study in which we investigate the physical properties of the boundary of a large low-latitude coronal hole. Differential Emission Measure analysis is used to derive the plasma properties of these regions. We also apply correlation dimension mapping analysis to measure the irregularities of the coronal hole boundary. We find that the leading boundary of this coronal hole has a slightly higher average plasma temperature, is associated with a stronger and more unipolar magnetic field, and has a smoother boundary line than the trailing counterpart. These differences are hypothesised to be direct consequences of the local magnetic field configurations of the coronal hole boundary: the leading boundary corresponds to large, well-organised coronal loops, and the trailing boundary corresponds to more dispersed, randomly orientated small magnetic bipoles. Hence, we suggest that the surrounding magnetic field structure and the nature of magnetic reconnection influence the properties of coronal hole boundaries.

Modern measurements in flavour physics rely on accurate simulations of signal and background processes, provided by a wide range of general-purpose and specialised Monte-Carlo event generators. Due to the inclusion of a larger amount of specialised decays of heavy hadrons, EvtGen is often the tool of choice for many scenarios. We investigate the phase-space sampling algorithm of EvtGen and demonstrate that it generates unphysical features in kinematic distributions of semileptonic $B$ decays involving resonances, originating from neglected phase-space factors. We provide a short-term solution to correct the affected simulated samples through reweighting of the hadronic invariant mass distribution.

We give the log $p$-adic relative monodromy-weight conjecture and prove it in certain cases.

As a powerful theoretical construct, the entanglement Hamiltonian (EH) encapsulates the essential entanglement properties of a quantum many-body system. From the EH, one can extract a variety of entanglement quantities, such as entanglement entropies, negativity, and the entanglement spectrum. However, its general analytical form remains largely unknown. While the Bisognano-Wichmann theorem gives an exact EH form for Lorentz-invariant field theories, its validity on lattice systems is limited, especially when Lorentz invariance is absent. In this work, we propose a general scheme based on the lattice-Bisognano-Wichmann (LBW) ansatz and multi-replica-trick quantum Monte Carlo methods to numerically reconstruct the entanglement Hamiltonian in two-dimensional systems and systematically explore its applicability to systems without translational invariance, going beyond the original scope of the primordial Bisognano-Wichmann theorem. Various quantum phases--including gapped and gapless phases, critical points, and phases with either discrete or continuous symmetry breaking--are investigated, demonstrating the versatility of our method in reconstructing entanglement Hamiltonians. Furthermore, we find that when the entanglement boundary of a system is ordinary (i.e., free from surface anomalies), the LBW ansatz provides an accurate approximation well beyond Lorentz-invariant cases. Our work thus establishes a general framework for investigating the analytical structure of entanglement in the complex quantum many-body systems.

We firstly carry out a systematic analysis on the spin $\frac{1}{2}^{+}\rightarrow\frac{3}{2}^{+}$ weak transition process in the framework of three-point QCD sum rules, where the initial and final states are doubly and singly charmed baryons. In the phenomenological side, all possible couplings of interpolating current to hadronic states are considered. In doing operator production expansion at QCD side, the contributions of the perturbative part, vacuum condensate terms of $\langle{\bar qq}\rangle$, $\langle g_{s}^{2}GG\rangle$, $\langle \bar q g_{s}σGq\rangle$ and $g_{s}^{2}\langle{\bar qq}\rangle^{2}$ are all considered. After the form factors in space-like region ($Q^2>0$) are obtained, the numerical results are extrapolated into time-like region ($Q^2<0$) by a fitting function. Using the predicted form factors, we finally analyze the semileptonic decays of $Ξ_{cc}^{++}\rightarrow Σ_{c}^{*+}l^{+}ν_{l}$, $Ξ_{cc}^{++}\rightarrow Ξ_{c}^{\prime*+}l^{+}ν_{l}$, $Ω_{cc}^{+}\rightarrowΞ_{c}^{\prime*0}l^{+}ν_{l}$ and $Ω_{cc}^{+}\rightarrow Ω_{c}^{*0}l^{+}ν_{l}$ with $l=e,μ$. The predictions in this work can deepen our understanding of the dynamics in the decay processes of doubly heavy baryons and provide useful information to explore the possibility of new physics in heavy baryonic decay channels.

RF sensing exploits phase-sensitive measurements of stray electromagnetic (EM) fields from wireless devices across various frequency bands to detect EM blockage and to reconstruct and map the surrounding environment in 2D/3D. Although blockage effects caused by objects or human motion are well-studied in ISM bands and frequencies up to 60~GHz, there is a significant lack of research for frequencies above 100~GHz. The paper proposes a unified signal processing framework for RF sensing in the sub-THz D-band (105--175~GHz), explicitly integrating EM blockage and scattering as a single process through the birth-death dynamics of multipath components (MPCs). The framework extracts, associates, and classifies MPCs from angle-delay measurements using statistically grounded detection and classification, enabling human-scale sensing from a single radio link. The modeling and classification of MPCs, along with large-scale EM parameters, are demonstrated through an indoor measurement campaign using multiple test targets. Experimental results show that newly formed, attenuated, and suppressed MPCs can be reliably identified with millimeter-scale delay resolution. Static object localization achieves average positioning errors of $8-20$~cm depending on range and material, while passive human localization yields errors of 12-17cm at 0.5m and 26-30cm at 2m, respectively. The proposed framework demonstrates that accurate sensing and localization are feasible at sub-THz frequencies using a single link.

[Abridged] The primary gravitational waves (PGWs) are generated in the early universe from the quantum vacuum during inflation. In slow roll inflation, the power spectrum (PS) of PGWs over large scales, which leave the Hubble radius during inflation, is nearly scale-invariant. However, over very small scales, which never leave the Hubble radius, the PS of PGWs behaves as k^2, where k denotes the wave number. We examine the PS of PGWs at such high wave numbers or frequencies when the PGWs are evolved post-inflation, through the epochs of radiation and matter domination. Firstly, we argue that the PS has to be regularized in order to truncate the unphysical k^2 rise at high frequencies. Assuming instantaneous transitions from inflation to the epochs of radiation and matter domination, we carry out the method of adiabatic regularization to arrive at the PS of PGWs over a wide range of frequencies. We show that the process of regularization truncates the k^2 rise and the PS of PGWs oscillates with a fixed amplitude about a vanishing mean value over small scales or, equivalently, at high frequencies. Secondly, we smooth the transition from inflation to radiation domination (to be precise, we smooth the 'effective potential' governing the equation of motion of PGWs) and examine the impact of the smoothing on the regularized PS of PGWs. With the help of a linear smoothing function, we explicitly show that the smoother transition leads to a power-law suppression in the amplitude of the oscillations (about the zero mean value) of the regularized PS of PGWs over small scales that never leave the Hubble radius during inflation. Our analysis indicates that, when transitions are involved, regularization as well as smooth transitions seem essential to ensure that the correlation functions of the PGWs in real space are well behaved. We discuss the directions in which our results need to be extended.

This paper presents a comprehensive dataset of doctoral theses defended in France between 1985 and 2025, constructed from multiple national academic metadata sources. The dataset is primarily based on data from the French national thesis platform and is enriched using additional authority and bibliographic databases to improve data quality, completeness, and interoperability. The data production pipeline includes the aggregation of heterogeneous sources, the correction of inconsistent identifiers, the enrichment of person and institution records, and the construction of derived variables describing academic careers, jury participation, institutional affiliations, and thesis characteristics. Additional identifiers from major academic repositories and library catalogues are integrated to facilitate linkage with external data sources and future dataset extensions. The resulting dataset provides structured information at the thesis, individual, and institutional levels, enabling both descriptive and relational analyses. This resource is particularly suited for research on doctoral education, academic networks, supervision practices, jury composition, institutional collaboration, and the evolution of research communities over time. The paper documents the data sources, processing pipeline, feature construction, data quality issues, and limitations, with the objective of facilitating reuse of the dataset by other researchers and supporting future extensions and longitudinal analyses of the academic system.

Percutaneous epicardial access (PEA), performed on a beating heart under fluoroscopy, enables arrhythmia treatment. However, advancing a needle toward the thin and moving pericardium remains highly challenging and risky. To address this problem, we present a physics-driven sonification method for Extended Reality (XR)-based multisensory navigation to enhance user perception during the critical needle landing phase in PEA. Dynamic cardiac anatomy from 4D CTA was reconstructed and registered to a real-world coordinate system. Real-time needle tracking provided the position of the needle tip relative to moving cardiac structures and drove an audio-visual feedback module. The visual display presented navigational cues and dynamic anatomy, while the auditory display encoded physiological cardiac states using a multilayer physical membrane model. A phantom study was conducted with twelve cardiologists performing needle insertions under visual-only and multisensory feedback. The multisensory method significantly improved navigation safety ($χ^2 = 11.30$, $p < 0.01$), reducing myocardial contact (3.64% vs. 7.27%) and increasing correct access (90.91% vs. 52.73%). Needle placement accuracy improved, with closer membrane proximity (Cliff delta = 0.19) and reduced variability ($p < 0.05$). Execution time was comparable, while time-accuracy correlations differed significantly between modalities ($p < 0.01$). NASA-TLX indicated lower cognitive load with multisensory guidance ($p < 0.01$). These results demonstrate the feasibility of physics-driven sonification for improving spatiotemporal awareness and supporting user-centered surgical navigation.

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then unless $D$ contains the complete symmetric directed graph on $d + 1$ vertices, $D$ admits a $μ$-measurable $d$-dicoloring with respect to any Borel probability measure $μ$ on $X$, and $D$ admits a $τ$-Baire-measurable $d$-dicoloring with respect to any Polish topology $τ$ compatible with the Borel structure on $X$. We also prove a definable version of Gallai's theorem on list dicolorings for directed graphs by showing that any Borel directed graph of bounded degree whose connected components are not Gallai trees is Borel degree-list-dicolorable.

Ptychographic reconstructions in reflection geometries are commonly interpreted with the same two-dimensional thin-sample model used in transmission, yet the validity of this approximation has not been established. We develop a three-dimensional weak-scattering description of reflection ptychography and derive explicit thickness criteria for when a two-dimensional model remains accurate. Because the sampled axial spatial frequency range is dominated by the rotation of the Ewald sphere rather than its curvature, reflection geometries impose far stricter thin-sample conditions than transmission geometries. The allowable thickness is reduced by one to two orders of magnitude for a representative extreme ultraviolet geometry, depending on the tolerance for appearance of artifacts. Simulations verify that conventional two-dimensional reconstructions may exhibit the thickness-dependent artifacts as predicted by the theory, with particularly strong distortions near specular Bragg minima. We further show that incorporating the correct depth-dependent propagation into the forward model resolves these distortions and enables recovery of sample thickness. These results establish practical validity limits for two-dimensional reflection ptychography and identify a path toward quantitative depth-sensitive reconstructions at all geometries.

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in $\mathbb{R}^3$.

The simulation of extensive air showers and particle cascades in general is a cornerstone of modern astroparticle physics. For more than two decades, CORSIKA, currently in version 7, has been one of the most widely used tools for this purpose. However, its architecture reflects design constraints of an earlier computing era, as well as increasingly limiting extensibility, maintainability, and adaptability to modern experimental requirements. CORSIKA 8 is a complete redesign of the original CORSIKA code, implemented in modern C++ and based on contemporary software engineering principles. It introduces a modular and extensible simulation framework with explicit handling of units, flexible geometry, and environment descriptions. In this paper, we present the design philosophy and core architecture of CORSIKA 8, describe the implementation of electromagnetic and hadronic shower physics, and validate air shower simulations against CORSIKA 7. The results demonstrate good agreement at the few-percent level for key observables, confirming the physics fidelity of CORSIKA 8. We also showcase new use cases that were beyond the capabilities of version 7, such as the simulation of cross-media showers and particle cascades in ice, including radio-signal propagation

We study a scalar-tensor extension of gravity with two scalar fields coupled to the Aharonov-Bohm extension of electrodynamics, where the scalar mode $S\equiv\partial_μA^μ$ is dynamical. In this framework the trace of the electromagnetic energy-momentum tensor is nonvanishing and the scalar $S$ induces an electro-gravitational coupling that can be enhanced by the vacuum expectation value of the second gravitational scalar. For bosonic matter described by a macroscopic wavefunction (as in superconductors), the coupling to the electromagnetic potential generates $S$ already at the semiclassical level, implying sizable junction-induced discontinuities. Including the scalar-tensor sector yields a nonlinear system for $S$ and a gravitational scalar combination $β$ that admits a bulk saturation solution $S_{\rm sat}^2=(Λλ_L^2)^{-1}$ and a corresponding threshold condition for macroscopic effects. We apply these results to pulsed discharges across normal-superconducting junctions and obtain scaling relations for the onset of anomalous gravitational signals in terms of current density, pulse duration, and superconducting volume, consistent with reported threshold behavior in two independent experimental configurations for a single microscopic parameter. We also present time-dependent propagating solutions in the weak-field regime and derive a class of one-dimensional traveling exact solutions of the nonlinear vacuum Einstein equations.

In June 2023, multiple pulsar timing array (PTA) collaborations provided evidence for the existence of a stochastic gravitational-wave background (SGWB). As a significant source of the SGWBs, scalar-induced gravitational waves (SIGWs) receive extensive attention. We explore the influence of anisotropic primordial power spectra on second-order SIGWs and derive explicit expressions for the energy density spectra. For specific anisotropic inflation models, we analyze the impacts of Finslerian inflation and gauge field inflation models on PTA and the Laser Interferometer Space Antenna (LISA) and generalize the findings to model-independent scenarios. Our results indicate that current PTA observations cannot rule out the existence of small-scale anisotropic primordial perturbations.

Since the publication of our review article Hyperspectral imaging solutions for brain tissue metabolic and hemodynamic monitoring: past, current and future developments in 2018, the technological and applicational landscape of the use of hyperspectral imaging (HSI) in brain sciences has evolved and transformed significantly. The number of studies and works where HSI has been deployed in its many forms to map and monitor the haemodynamic and metabolic states of cerebral tissues have grown exponentially, to such a point where an update on the cur-rent state of the art is timely, and we believe would be desirable for both long-term experts in the field, as well as for any new researcher approaching it for the first time. In this commentary, we provide a renewed perspective on the newest and latest developments in brain haemodynamic and metabolic monitoring with HSI over the past eight years. Our hope is that even greater breakthroughs and broader, more numerous novel applications will come forward in the future for the technology, that may benefit from this new overview, as they did from the original one.

Starting from generalized position operators, we derive complex and quaternionic angular momentum operators along with their commutation algebra as well. These algebras differ from the standard Hermitian ones, especially in terms of commutation relations involving partial and total angular momentum operators. Despite these differences, the effective quantum expectation values obtained from slightly deformed algebras align with those from the conventional Hermitian algebra. This suggests that even though the wave functions and resulting dynamics differ from standard quantum Hermitian behavior, these deformed algebras can still be effectively understood as valid angular momentum algebras.

This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of characteristic different from two and three. We also compare our lists with the most recent classifications over the complex numbers and with the nilpotent case over arbitrary base fields in dimension three, adding some comments.

Latin America's education systems are fragmented and segregated, with substantial differences by school type. The concept of school efficiency (the ability of school to produce the maximum level of outputs given available resources) is policy relevant due to scarcity of resources in the region. Knowing whether private and public schools are making an efficient use of resources --and which are the leading drivers of efficiency-- is critical, even more so after the learning crisis brought by the COVID-19 pandemic. In this paper, relying on data of 2,034 schools and nine Latin American countries from PISA 2022, I offer new evidence on school efficiency (both on cognitive and non-cognitive dimensions) using Data Envelopment Analysis (DEA) by school type and, then, I estimate efficiency leading determinants through interpretable machine learning methods (IML). This hybrid DEA-IML approach allows to accommodate the issue of big data (jointly assessing several determinants of school efficiency). I find a cognitive efficiency gap of nearly 0.10 favouring private schools and of 0.045 for non-cognitive outcomes, and with a lower heterogeneity in private than public schools. For cognitive efficiency, leading determinants for the chance of a private school of being highly efficient are higher stock of books and PCs at home, lack of engagement in paid work and school's high autonomy; whereas low-efficient public schools are shaped by poor school climate, large rates of repetition, truancy and intensity of paid work, few books at home and increasing barriers for homework during the pandemic.

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria, which we then demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.

Embedding non-Markovian open quantum dynamics into an enlarged Markovian space offers a powerful route to nonperturbative simulations, where the dynamics of the extended space can be governed by multiple distinct Markovian equations. We show that these distinct embeddings arise from different unravelings of Gaussian bath self-energies, generating a family of deterministic, time-local equations for the extended system. Using the Brownian-oscillator spectral density as an illustrative example, we clarify the relationships among existing approaches, including the Hierarchical Equations of Motion (HEOM) and the Lindblad--pseudomode formalism, and demonstrate how this framework enables numerically stable and efficient simulations. This work provides both a transparent theoretical foundation for embedding techniques and a flexible platform for developing new methods to simulate non-Markovian quantum dynamics.

Galactic Cosmic Rays (GCRs) have to travel through the heliosphere before they interact with the Earth's atmosphere. During this, they are deflected by the Sun's magnetic field, causing variations in this field to imprint on the flux, spectrum and angular distribution of GCRs detected at or near Earth. Studies of these variations over the past several decades have revealed the impact of both transient phenomena such as solar flares, coronal holes, sunspot activity and coronal mass ejections (CMEs) as well as their effects such as Forbush Decreases (FDs), precursors and Ground-Level Enhancements (GLEs). Periodic variations, such as due to the solar diurnal modulation, the 27-day solar rotation, the 11-year solar cycle, and the 22-year solar magnetic cycle have also been characterized. These Sun-induced phenomena are most prominent in GCR intensity variations up to $\sim$30 GeV/nuc, beyond which the influence of solar modulation decreases rapidly as the gyro-radii of GCRs exceed the characteristic size of the heliosphere ($\sim$100 AU).

Baryon Acoustic Oscillation (BAO) measurements play a key role in ruling out post-recombination solutions to the Hubble tension. However, because the data compression leading to these measurements assumes a fiducial $Λ$CDM cosmology, their reliability in testing late-time modifications to $Λ$CDM has at times been called into question. We play devil's advocate and posit that fiducial cosmology assumptions do indeed affect BAO measurements in such a way that low-redshift acoustic angular scales (proportional to the Hubble constant $H_0$) are biased low, and test whether such a rescaling can rescue post-recombination solutions. The answer is no. Firstly, strong constraints on the shape of the $z \lesssim 2$ expansion history from unanchored Type Ia Supernovae (SNeIa) prevent large deviations from $Λ$CDM. In addition, unless $Ω_m$ is significantly lower than $0.3$, the rescaled BAO measurements would be in strong tension with geometrical information from the Cosmic Microwave Background. We demonstrate this explicitly on several dark energy (DE) models ($w$CDM, CPL DE, phenomenologically emergent DE, holographic DE, $Λ_s$CDM, and the negative cosmological constant model), finding that none can address the Hubble tension once unanchored SNeIa are included. We argue that the $Λ_s$CDM sign-switching cosmological constant model possesses interesting features which make it the least unpromising one among those tested. Our results demonstrate that possible fiducial cosmology-induced BAO biases cannot be invoked as loopholes to the Hubble tension "no-go theorem", and highlight the extremely important but so far underappreciated role of unanchored SNeIa in ruling out post-recombination solutions.

We prove the existence of multisoliton solutions of the three-dimensional gravitational Hartree equation whose trajectories follow many body dynamics of hyperbolic, parabolic or hyperbolic-parabolic types. This work generalizes and improves the result of Krieger-Martel-Raphaël on two-soliton solutions.

In recent years, strong-field processes such as high-order harmonic generation (HHG) and above-threshold ionization driven by nonclassical states of light have become an increasingly popular field of study. The theoretical modeling of these processes often applies an approximate phase-space expansion of the nonclassical driving field in terms of coherent states, which has been shown to accurately predict the harmonic spectrum. However, its accuracy for the computation of quantum optical observables like the degree of squeezing and photon statistics has not been thoroughly considered. In this work, we introduce this approximative phase-space description and discuss its accuracy, and we find that it mischaracterizes the quantum optical properties of the driving laser by making it an incoherent mixture of classical states. We further show that this error in the driving field description maps onto the light emitted from HHG, as neither sub-Poissonian photon statistics nor quadrature squeezing below vacuum fluctuations can be captured by the approximative phase-space description. Lastly, to benchmark the approximative phase-space description, we consider the quantum HHG from a one-band model, which yields an exact analytical solution. Using the approximative phase-space representation with this specific model, we find a small quantitative error in the quadrature variance of the emitted field that scales with pulse duration and emitter density. Our results show that using this approximative phase-space description can mischaracterize quantum optical observables. Attributing physical meaning to such results should therefore be accompanied by a quantitative analysis of the error.

We undertake a Shannon theoretic study of the problem of communicating bit streams over a 3-user classical-quantum interference channel (3-CQIC) and focus on characterizing inner bounds. We design a new coding strategy based on (i) coset codes possessing algebraic closure properties and (ii) decoding POVMs to decode bi-variate interference efficiently. Needing to perform simultaneous decoding, we enhance Sen's powerful technique of tilting, smoothing, and augmentation - originally designed only for IID codes - to decode into `functions of codebooks'. Developing analysis techniques to combine all of these elements, we derive a new inner bound to the capacity region of a 3-CQIC. The derived inner bound subsumes all currently known inner bounds and is analytically proven to be strictly larger for identified examples, including non-commutative `additive' and `non-additive' ones.

This summary of the doctoral thesis provides a comprehensive formulation of the Extended Discrete Fourier Transform (EDFT), derived directly from the Fourier integral and its orthogonality properties. The method is obtained by solving weighted least-squares estimators in both continuous and discrete domains, yielding an adaptive frequency-domain representation that remains fully consistent with the classical Fourier framework. In the special case of uniformly sampled data on a uniform frequency grid of the same size, the EDFT reduces exactly to the classical Discrete Fourier Transform (DFT). However, when the analysis grid exceeds the number of observed samples, EDFT circumvents conventional zero-padding by optimizing the transformation basis over the extended frequency set. This enables accurate spectral estimation from incomplete or nonuniformly sampled data. Consequently, the EDFT achieves enhanced frequency resolution in regions of strong spectral content while maintaining global resolution balance, thereby remaining consistent with the uncertainty principle. The inverse EDFT reconstructs the original signal and produces extrapolated or interpolated samples wherever spectral information is available. The EDFT requires no explicit separation of deterministic and stochastic components and accurately captures broadband, transient, and sinusoidal features simultaneously. Simulation studies confirm its robustness under nonuniform sampling, multiple Nyquist zones, missing-data conditions, and signals with mixed spectra comprising both line and continuous components. Although iterative computation of the EDFT entails higher numerical cost compared to the classical DFT, this limitation - significant in the 1990s - has been largely mitigated by modern computational resources, rendering the EDFT practical for contemporary signal analysis applications.

We report a three-dimensional lumped-element multimode microwave resonator that enables homogeneous collective manipulation and dispersive readout of a macroscopic spin ensemble. By exploiting geometric symmetry, two antisymmetric modes with strongly suppressed cross-talk are engineered to spatially overlap and couple to the same ensemble at distinct frequencies. Using negatively charged nitrogen-vacancy centers in diamond at 28 mK, we observe collective strong coupling with a coupling strength of 5.0 MHz and demonstrate non-destructive dispersive readout via a detuned mode. The compact design, tunable coupling, and high field homogeneity make this resonator a versatile device for hybrid spin-photon systems and multimode solid-state quantum technologies.

In this paper, we investigate the approximate analytical bound states with a linear combination of two diatomic molecule potentials, Yukawa and four parameters potentials, within the framework of the path integral formalism. With the help of an appropriate approximation to evaluate the centrifugal term, the energy spectrum and the normalized wave functions of the bound states are derived from the poles of Green's function and its residues. The partition function and other thermodynamic properties were obtained using the compact form of the energy equation.