We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.

We propose a new parameter called proofdoor in an attempt to explain the efficiency of CDCL SAT solvers over formulas derived from circuit (esp., arithmetic) verification applications. Informally, given an unsatisfiable CNF formula F over n variables, a proofdoor decomposition consists of a chunking of the clauses into A1, . . . , Ak together with a sequence of interpolants connecting these chunks. Intuitively, a proofdoor captures the idea that an unsatisfiable formula can be refuted by reasoning chunk by chunk, while maintaining only a summary of the information (i.e., interpolants) gained so far for subsequent reasoning steps. We prove several theorems in support of the proposition that proofdoors can explain the efficiency of CDCL solvers for some class of circuit verification problems. First, we show that formulas with small proofdoors (i.e., where each interpolant is O(n) sized, each chunk Ai has small pathwidth, and each interpolant clause has at most O(log(n)) backward dependency on the previous interpolant) have short resolution (Res) proofs. Second, we show that certain configurations of CDCL solvers can compute such proofs in time polynomial in n. Third, we show that commutativity (miter) formulas over floating-point addition have small proofdoors and hence short Res proofs, even though they have large pathwidth. Fourth, we characterize the limits of the proofdoor framework by connecting proofdoors to the partially ordered resolution proof system: we show that a poor decomposition of arithmetic miter instances can force exponentially large partially ordered resolution proofs, even when a different decomposition (i.e., small proofdoors) permits short proofs.

We investigate the magnetization dynamics of $TmFeO_3$ single crystals across the spin-reorientation phase transition using broadband microwave absorption spectroscopy up to 87.5 GHz. Temperature- and magnetic-field-dependent antiferromagnetic resonance measurements reveal the characteristic softening of the quasi-ferromagnetic (q-FM) resonance mode at the $Γ_2\rightarrowΓ_{24}$ and $Γ_{24}\rightarrowΓ_4$ transition points. The finite magnon gap observed at the transition points reflects the strong magnetoelastic coupling. In addition to the uniform q-FM mode, multiple magnon modes appear in the intermediate $Γ_{24}$ phase, separated by approximately 0.5--2 GHz and exhibiting similar field and temperature dependence. These additional modes are attributed to nonuniform spin-wave excitations arising from the periodic magnetic domain structure present in the intermediate phase and their hybridization with acoustic phonons mediated by strong magnetoelastic coupling. Our results demonstrate that the spin-reorientation transition in $TmFeO_3$ provides a natural platform for generating multiple hybridized magnon modes, offering new opportunities for tunable magnonic excitations in rare-earth orthoferrites.

We consider spin-orbit coupled Bose-Einstein condensates with cubic-quintic nonlinear interaction within the framework of second quantization formulation and find eigen states using numerical simulation and mean-field approximation. We show that two low-lying Schrodinger cat states remain degenerate up to a certain value of Raman coupling strength and these states can serve as qubit basis. We take three different nonlinear perturbations and find that the perturbations can result in different rotations of qubit state on Bloch sphere. We calculate the unitary operator corresponding to each perturbation and suggest the possibilities for obtaining various gates in ultra-cold atomic system.

Controlling physical responses through symmetry breaking is a central paradigm in quantum materials, enabling novel functionalities. Here we determine the effects of spin-group-symmetry breaking on nonlinear optical responses of collinear altermagnetic insulators. Using shear strain as an example, we show that the direction of symmetry-breaking induced components of charge and spin photocurrents are locked to the sign of the strain. In the absence of spin-orbit coupling, this effect is intuitively captured by the spin-gap asymmetry--an imbalance between spin-up and spin-down direct band gaps which couples trilinearly with the Néel order and the strain. We demonstrate this mechanism with density functional theory calculations on the recently proposed altermagnet CuWP$_2$S$_6$. Having symmetry-guided control of both charge and spin photocurrents allows, vice versa, to reveal and investigate altermagnetism in insulating materials by exploration of their optical responses.

Two-dimensional Coulomb gases on an annulus at a special inverse temperature $β= 2$ are studied by using the orthogonal polynomial method borrowed from the theory of random matrices. The correlation functions among the Coulomb gas molecules are written in determinant forms and their asymptotic forms in the thermodynamic limit are evaluated. When the Coulomb gas system has a continuous rotational symmetry, the corresponding orthogonal polynomials are monomials, and one can see a universal behavior of the correlation functions in a thin annulus limit. In a system with a discrete rotational symmetry, the corresponding orthogonal polynomials are not in general monomials, and a breakdown of the universality is observed.

In this work we show that in general there is no Courant-like bound for Neumann domain count. In order to do that we construct a sequence of domains $Ω^n$ such that the first Dirichlet eigenfunction for $Ω^n$ has at least $n$ Neumann domains. Also a special case of convex domains is considered and sufficient conditions for existence of Courant-like bound for small eigenvalues are found.

A large-scale quantum circuit can be partitioned into multiple subcircuits through circuit cutting, where each subcircuit is executed multiple times and the expectation value of the original circuit is reconstructed by classical post-processing from their measurement (sampling) results. In this process, appropriate cut locations are identified after the user-designed quantum circuit, including multi-qubit gates that act on three or more qubits, has been decomposed into single-qubit gates and two-qubit gates such as the CNOT gate. Here, we present a method for reducing the sampling overhead, which refers to the increase in the number of samples required due to the cutting process, by modifying the decomposition strategy of multi-qubit gates. Using MCX and CCCX gates as representatives of multi-qubit gates, we demonstrate that the proposed decomposition method, which introduces a small number of ancilla qubits according to the identified cut locations, effectively decreases the sampling overhead.

The availability of generative Artificial Intelligence (AI) tools such as ChatGPT or GitHub Copilot is reshaping the way in which software is developed, evolved, and maintained. Oftentimes, developers leave traces of such an usage in software artifacts. This allows not only to understand how AI is used in software development, but also to let others be aware how such software artifacts were created, e.g., for licensing or trustworthiness purposes. This paper-building upon our preliminary work presented at MSR 2024-aims at qualitatively investigating on the self-admitted use of two very popular generative AI tools - ChatGPT and GitHub Copilot - in software development. To this aim, we mined GitHub for such traces, by looking at commits, issues and pull requests (PRs). Then, through a manual coding, we create a taxonomy of 64 different ChatGPT and GitHub Copilot usage tasks, grouped into 7 categories. By repeating our previous analysis two years after and by extending it to GitHub Copilot, we show how the usage avenues have been expanded, the extent to which developers perceived such a generative AI usage useful, and whether some concerns occurring more than one year ago are no longer present. The taxonomy of tasks we derived from such a qualitative study provided (i) developers with valuable insights into how generative AI can be integrated into their workflows, and (ii) researchers with a clear overview of tasks that developers perceive as well-suited for automation.

The impulsive phase of a solar flare is known to generate strong turbulence and to transfer magnetic energy into accelerated electrons. Recognizing the importance of angular diffusion on the dynamics of the accelerated electrons, we extend previous treatments by deriving analytic solutions for the electron flux and associated energy deposition in two regimes: scattering dominated by inelastic Coulomb collisions and scattering dominated by elastic interactions with turbulent scattering centers. We show that the turbulence-dominated scattering term strongly reshapes the spatial distribution of the plasma heating: compared to the traditional collisional thick-target approach, turbulent scattering could lead to an order-of-magnitude increase in coronal heating and an even greater suppression of chromospheric heating. Scattering also acts to reduce the anisotropy of the electron distribution and so reduces the net current associated with the nonthermal electrons. The return-current Ohmic heating is accordingly reduced to a level that renders it negligible compared to direct collisional heating. The results have significant implications for models of atmospheric response to impulsive phase energy release, in particular chromospheric evaporation, flare-driven coronal heating, the formation of loop-top hard X-ray sources, and the longstanding discrepancy between modeled and observed soft X-ray line profiles.

The transient time correlation function (TTCF) method is widely used in molecular fluids to compute non-equilibrium transport quantities, providing improved signal-to-noise ratios in ensemble averages without requiring prohibitively large sample sizes. In spite of its success in molecular and turbulent fluid systems, the method has not been systematically explored for more general non-equilibrium dynamical systems, including geophysical applications where the invariant measure is typically unknown. In this work, we present an analytical and numerical investigation of the TTCF method for computing nonlinear response functions in systems far from equilibrium. We discuss its relation to the spectral theory of stochastic systems, highlighting regimes where linear theory is insufficient and the advantages of TTCF. The aim of this work is to provide a framework for studying transient and steady-state responses using the TTCF approach in a broad class of nonequilibrium systems.

Colloidal plasmonic-photonic crystals represent a class of hybrid materials composed of a dielectric colloidal spheres photonic lattice and a metal plasmonic film. In this work, the optical properties of a linear array colloidal plasmonic-photonic crystal consisting of silver films deposited over linear arrays of polystyrene microspheres are analysed in detail. Experimental and simulated optical transmittance and reflectance spectra both with unpolarized and polarized light are used to investigate the optical response of the linear plasmonic-photonic crystal. Among the various photonic/plasmonic modes observed, the existence of both propagative plasmonic-photonic hybrid mode and localized surface plasmon mode can be mentioned. The spectral tunability of these structures is highlighted by studying the dependence of the optical response on geometrical parameters such as sphere diameter and grating period. Finally, the linear plasmonic-photonic crystal exhibits a polarization-selective surface-enhanced Raman scattering effect, making them of interest for both fundamental studies and development of applications based on surface-enhanced Raman spectroscopy or surface-enhanced fluorescence.

We propose a simple yet effective local smoothness indicator for the downwind stencil in central-upwind weighted essentially non-oscillatory (WENO) schemes of even order for hyperbolic conservation laws. Starting from an odd-order upwind WENO scheme, we construct an even-number-of-points stencil by incorporating a downwind substencil whose smoothness indicator is the arithmetic mean of all local smoothness indicators. This straightforward averaging approach incorporates regularity information from the entire stencil without requiring additional tuning parameters or complex formulations. Combined with affine-invariant Z-type nonlinear weights and a carefully designed global smoothness indicator, the resulting scheme, termed WENO-ZA6 for the sixth-order case, achieves optimal convergence rates at critical points up to second order, exhibits favorable dispersion and dissipation properties as confirmed by approximate dispersion relation analysis, and provides sharp, essentially non-oscillatory resolution of discontinuities. Numerical experiments on scalar problems and the one- and two-dimensional Euler equations demonstrate that WENO-ZA6 achieves accuracy comparable to or better than existing sixth-order central-upwind schemes (WENO-CU6, WENO-S6) and the seventh-order WENO-Z7, while requiring approximately 15\%--21\% less computational time. The framework extends naturally to fourth-, eighth-, and tenth-order schemes.

In two-dimensional systems with space-time inversion symmetry, such as $C_{2z}T$, the reality condition on wave functions gives rise to real band topology characterized by the Euler class, a $\mathbb{Z}$-valued topological invariant for a pair of real bands in the Brillouin zone. In this paper, we study three-dimensional $C_{2z}T$-symmetric insulators characterized by $\bar{e}_2$, defined as the difference in the Euler classes between two $C_{2z}T$-invariant planes in the three-dimensional Brillouin zone. By deriving effective surface Hamiltonians from generic low-energy continuum Hamiltonians characterized by the topological invariant $\bar{e}_2$, we reveal that multiple gapless boundary states exist at the domain walls of the surface mass, which give rise to the multiple chiral hinge modes. We also show that three-dimensional insulators characterized by $\bar{e}_2=N$ support $N$ chiral hinge modes. Notably, due to the constraint of two occupied bands in our system, these phases are distinct from stacked Chern insulators composed of $N$ layers. Furthermore, we construct tight-binding models for $\bar{e}_2=2$ and $3$ and numerically demonstrate the emergence of two and three chiral hinge modes, respectively. These results are consistent with those obtained from the surface theory.

Current approaches for knowledge graph construction with RML focus on full RDF graph materialization without considering user queries. As a result, mapping engines are inefficient in dynamic query environments, materializing large graphs even when only a small subset is needed to answer user queries. In this paper, we formally define satisfiability for SPARQL queries with respect to RDF data obtained via RML mappings and use this property to prune RML mappings for partial RDF graph materialization. Evaluation on the GTFS-Madrid benchmark shows that pruning significantly reduces materialization time, and RDF graph size while also noticeably improving querying time. Thus, enabling existing materialization engines to efficiently support generating RDF graphs in dynamic federated querying environment where user queries change frequently.

In studies of bundled modalities, we encode a complex conceptual notion into the semantics of a single modal operator and study its logic. Although there is already a substantial body of work on various concrete bundled operators, we still lack a general understanding of them. In this paper, we provide a general theory of the expressivity and axiomatization of bundled modalities. We offer a uniform way to define bisimulations for arbitrary bundled modalities and justify our definition by the corresponding Hennessy-Milner property. We also define a special class of bundled modalities called convex bundles. This class covers most bundled modalities studied in the literature, and their axiomatizations can be done with the help of convex neighborhood semantics and corresponding representation results. As case studies, we axiomatize the "someone knows" bundle $\bigvee_{a \in A} \Box_a ϕ$ over $S5$-models, the "disagreement in group" bundle $\bigvee_{a, b \in A} \Box_a ϕ\wedge \Box_b \neg ϕ$ over $KD45$-models, and the "belief without knowledge" bundle $B ϕ\wedge \neg K ϕ$ over $S4.2$-models.

This work analyzes the RAID dataset to evaluate human responses to affine image distortions, including rotation, translation, scaling, and Gaussian noise. Using Mean Squared Error (MSE), the study establishes human detection thresholds for these distortions, enabling comparison across types. Statistical analysis with ANOVA and Tukey Kramer tests reveals that observers are significantly more sensitive to Gaussian noise, which consistently produced the lowest detection thresholds. Fourier analysis further shows that high-frequency components act as a visual mask for Gaussian noise, demonstrating a strong correlation between high frequency energy and detection thresholds. Additionally, spectral orientation influences the perception of rotation. Finally, the study employs the PixelCNN model to show that image probability significantly correlates with detection thresholds for most distortions, suggesting that statistical likelihood affects human visual tolerance.

We apply to the nucleon-nucleus inelastic process a fully coherent microscopic multiple scattering approach. Our study addresses the complexities inherent in characterizing inelastic scattering events, offering a comprehensive theoretical model grounded in the reaction theory. The approach is based on the distorted-wave approximation and requires the knowledge of three potentials, which give the initial and final distorted wave functions and the transition potential. All of them are derived just like the microscopic optical potential for elastic nucleon-nucleus scattering we derived in previous papers of ours within the framework of the Watson multiple scattering theory and adopting the impulse approximation. The potentials are obtained by folding nonlocal ab initio nuclear densities from the No-Core Shell Model (NCSM) with a nucleon-nucleon $t$ matrix computed with a chiral interaction consistent with the one used in the calculation of the density. The only difference in the formal expressions of the three potentials resides in the nuclear density, where we use the ground and excited state densities of the target and the transition density. By extending methods traditionally applied to elastic scattering, we incorporate the effects of inelastic transitions enabling an accurate description of the experimental differential cross section. The predictive power of our numerical results is benchmarked against empirical data of inelastic proton scattering off $^{12}$C, for the transition to the $2^+$ state at 4.44 MeV, in a range of projectile energies of 65-300 MeV. The generally good description of the experimental cross sections as functions of the scattering angle gives clear evidence of the reliability and robustness of a model that does not contain any free adjustable parameters.

We construct static, spherically symmetric wormhole solutions with a nontrivial redshift function, inspired by noncommutative geometry, in which point sources are replaced by Gaussian smearing of minimal length, yielding a regular shape function. Within this framework, we derive model-independent relations that isolate the role of the redshift function in controlling the stress-energy components and the violation of the null energy condition (NEC). Negative or suitably tuned redshifts confine the exotic matter to a thin neighborhood of the throat. We then reformulate this redshift engineering in matter terms through a quasi-de Sitter equation of state (EOS) with localized Gaussian or Lorentzian perturbations, obtaining minimally exotic wormholes that are regular, horizon-free, and asymptotically flat. Finally, we extend the analysis to a Chaplygin-like EOS, introducing a nonlinear coupling between pressure and density that yields redshift wells with possible local blueshift regions and tunable anisotropies governed by a certain nonlinearity parameter. Together, these results provide a unified and physically transparent framework for constructing traversable noncommutative-geometry-inspired wormholes with controlled, spatially localized exotic matter content.

We provide simple necessary and sufficient conditions under which a path constitutes a solution to an infinite-horizon, continuous-time optimal control problem. We prove transversality conditions under standard assumptions. We also present some applications to economic models.

Electron spin resonance scanning tunneling microscopy (ESR-STM) has become a powerful tool for probing spin dynamics and coherence of individual atoms and molecules on surfaces. In this work, we perform Rabi oscillation and Hahn echo pulse protocols on individual iron phthalocyanine (FePc) molecules on MgO/Ag(001) using ESR-STM. While Hahn echo protocols are widely used to extract spin coherence times, we show that in ESR-STM they are particularly susceptible to misinterpretation due to tunneling electrons generated by the applied radio-frequency (RF) voltage. The RF voltage not only drives the spin, but simultaneously probes and relaxes it, which consequently leads to an exponential decay that reflects spin relaxation rather than intrinsic phase coherence. We moreover show that varying both delay times in the refocusing pulse sequence is a reliable way to ensure a coherent nature of the echo signal. The extracted decay for the latter protocol suggests that T2 is approximately 30 ns and is thus closer to the decoherence time observed in Rabi oscillation measurements. This is significantly shorter than values reported in previous echo measurements. Our findings underscore the need for caution in interpreting T2 times from Hahn echo and Carr-Purcell protocols in ESR-STM and provide practical criteria for distinguishing true spin echoes from tunneling-induced relaxometry signals.

Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $λ_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $λ_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $λ_2\le 4(D-1)/\ell$ to the optimal value~$1$. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $λ_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees.

Squeezed states of light are used for precision metrology and quantum-enhanced measurements, with applications spanning communication and sensing. State-of-the-art squeezed-light sources typically rely on optical cavities to achieve high, usable levels of squeezing. Recently, waveguide-based squeezed-light sources have demonstrated significant improvements in achievable squeezing, with performance currently limited by fabrication-induced losses. In this work, we present a detailed analysis of waveguide-based squeezers, examining the effects of phase noise, multiple loss mechanisms, and fundamental light leakage seeding the squeezer. We further investigate a cascaded squeezer architecture, in which a second waveguide operates as a phase-sensitive amplifier to mitigate out-coupling and detection losses. Owing to their ease of integration, robustness to high pump powers, and low intrinsic phase noise, we propose waveguide-based squeezed-light sources as a promising alternative for quantum noise reduction in future gravitational wave detectors, such as the Einstein Telescope.

Hydrogen-based magneto-ionics is a promising approach for rapid magnetoelectric control of spintronic devices. Most investigations so far into the magneto-ionic manipulation of rare-earth transition-metal alloys have used electrochemical methods for evaluating the magnetoelectric properties, but this technique makes it difficult to discriminate between the effects of competing ionic species. In this work, we use atmospheric loading to evaluate the effect of an isotope of hydrogen, deuterium, on the magnetic properties of TbCo films using in-situ polarized neutron reflectometry. With this approach, we are able to simultaneously measure the magnetization, thickness expansion and deuterium concentration of TbCo films. We quantitatively observe the deuterium concentrations at which the paramagnetic phase transition occurs for a Tb-rich film, and the weakening of out-of-plane magnetic anisotropy for a Co-rich film. For the Tb-rich film the expansion of the film thickness is the primary mechanism identified for the paramagnetic phase transition, while for the Co-rich film no thickness expansion is observed. We also find that an oxidized interface is insensitive to deuterium loading, but remains exchange coupled to the rest of the film and can be indirectly manipulated by loading of deuterium in the alloy. We expect these results to be directly translatable to that of hydrogen.

Metal-coated microsphere monolayers (MCM) are a class of plasmonic crystals consisting of noble metal films over arrays of self-assembled colloidal microspheres. Despite their ease of fabrication and tunable plasmonic response, their optical sensing potential has been scarcely explored. Here, silver coated polystyrene sphere monolayers are proposed as surface plasmon resonance sensors capable of functioning in both transmission (T) and reflection (R) readout modes. An original and key point is the use of ~200 nm colloids, smaller than in MCM studied before. It allowed us to reveal a previously unobserved, additional/secondary Enhanced Optical Transmission band, which can be exploited in sensing, with higher sensitivity than the better-known main transmission band. The reflection configuration however, is almost an order of magnitude more efficient for sensing than the transmission one. We also evidenced a strong impact of the adsorbate location on the metal surface on the sensing efficiency. Electric field distribution analysis is performed to explain these results. Proof-of-concept experiments on the detection of 11-MUA molecular monolayers, performed in both readout modes, confirm the behaviors observed through FDTD simulations. Results in this paper can serve as guidelines for designing optimized sensors based on metal-coated colloidal monolayers, and more generally for plasmonic sensors based on metal nanostructured films.

Structured light carrying orbital angular momentum enables new regimes of nonlinear light-matter interaction. Here we develop a molecular quantum electrodynamics description of third-harmonic generation (THG) driven by focused Laguerre-Gaussian beams in isotropic molecular media. We show that the nonparaxial longitudinal field components of a tightly focused beam permit THG with circularly polarized excitation in an isotropic fluid, a process forbidden for plane waves and paraxial beams. Within the electric-dipole approximation, the resulting emission is independent of the sign of the vortex charge. Including electric-magnetic dipole interference introduces a chiral contribution to the nonlinear response, giving rise to third-harmonic vortex dichroism (THVD). The emitted intensity then acquires a component linear in the topological charge \(\ell\), reversing sign with either the wavefront handedness or molecular chirality. Numerical modelling reveals corresponding spatial asymmetries in the harmonic field. These results establish both an allowed pathway for circularly polarized THG in isotropic fluids and the first chiroptical analogue of THG in such media, identifying orbital angular momentum as a new control parameter for nonlinear chiral spectroscopy.

The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.

We establish a pointwise limit theorem for a broad class of pa\-ra\-me\-ter-\-de\-pen\-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several existing results and yields non-distributional characterizations of Sobolev-type spaces, both in the scalar and in the vector-valued setting. More precisely, for any open set $Ω\subset \mathbb{R}^n$ and any $p\in (1, \infty)$, we provide a characterization of the Sobolev space $W^{1,p}(Ω; \mathbb{R}^m)$. In addition, we characterize the space $E^{1,p}(Ω;\mathbb{R}^n)$ of $L^p$ maps with $p$-integrable distributional symmetric gradient.\\ Finally, for all $p\in [1, \infty)$, we show that these seminorms converge to integral functionals with convex, $p$-homogeneous integrands associated with the distributional gradient and the symmetric gradient.

Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising solution for combinatorial optimization problems using a hybrid quantum-classical framework. Among combinatorial optimization problems, the Maximum Cut (Max-Cut) problem is particularly important due to its broad applicability in various domains. While QAOA-based Max-Cut solvers have been developed, they primarily favor solution accuracy over execution efficiency, which significantly limits their practicality for large-scale problems. To address the limitation, we propose ParaQAOA, a parallel divide-and-conquer QAOA framework that leverages parallel computing hardware to efficiently solve large Max-Cut problems. ParaQAOA significantly reduces runtime by partitioning large problems into subproblems and solving them in parallel while preserving solution quality. This design not only scales to graphs with tens of thousands of vertices but also provides tunable control over accuracy-efficiency trade-offs, making ParaQAOA adaptable to diverse performance requirements. Experimental results demonstrate that ParaQAOA achieves up to 1,600x speedup over state-of-the-art methods on Max-Cut problems with 400 vertices while maintaining solution accuracy within 2% of the best-known solutions. Furthermore, ParaQAOA solves a 16,000-vertex instance in 19 minutes, compared to over 13.6 days required by the best-known approach. These findings establish ParaQAOA as a practical and scalable framework for large-scale Max-Cut problems under stringent time constraints.

In this article, I trace the early historical developments that ultimately led to the creation of the atomic bomb. Even after the completion of weapons, many scientists continued to argue that nuclear armaments were indispensable for maintaining the global balance of political power [1]. This study focuses on several scientists who confronted profound moral dilemmas concerning the use of bombs against Japan. Some openly opposed its deployment. Others sought to warn a Japanese physicist in the hope of averting further devastation. Still, others expressed deep remorse in its aftermath. In addition, the experience of an individual directly affected by the bombing is discussed. By examining these episodes, this article aims to contribute to the ongoing discourse on how scientific research should be guided by ethical principles in the future.