Search Results (55)

Grid-based density clustering methods—FlowGrid, FLOPTICS, and grid-accelerated DBSCAN and OPTICS—partition a $d$-dimensional feature space into an $m^{\hspace{0.17em}d}$ array of bins and group the non-empty bins by density reachability. They leave a symmetry unexploited: relabelling and reflecting the feature axes permutes the bins while preserving density. We formalise this as an action of the hyperoctahedral group $B_d=C_2\wr S_d$ of order $2^{\hspace{0.17em}d}d!$ on bin-occupancy functions. We prove that the action preserves the Chebyshev bin-adjacency underlying grid reachability (Theorem 1); that grid density clustering is $B_d$-equivariant (Theorem 2); that configurations up to symmetry are counted by a Burnside average over $B_d$ (Theorem 3), with an explicit parity-split closed form for the octahedral case at every resolution, which we have not found recorded elsewhere (Theorem 4 and Corollary 2); and that a canonical-form algorithm computes a unique representative per orbit in $O\left(2^{\hspace{0.17em}d}d!\hspace{0.17em}{m}^{\hspace{0.17em}d}\right)$ time (Theorem 5). We are explicit about scope: a generic dataset has a trivial stabiliser, so there is no per-run gain; the benefit is canonical indexing, deduplication of symmetry-closed libraries, and caching when the same measurement recurs under a different axis convention. The group specialises to the $D_4$ and $O_h$ symmetries of the DR Code. Symmetry enters twice: as the group acting on the grid, and as the line between symmetric instances and the generic asymmetric ones. The finite instances and enumeration formulae are confirmed computationally; the general statements are established by proof. Full article

We analyse one-dimensional bound states of spin-zero particles coupled to external time-like vector potentials within the Feshbach–Villars representation. The study is organised around a practical criterion of physical admissibility that distinguishes genuine bound states from scattering solutions and finite-box artefacts by combining asymptotic decay, parity, matching conditions, conserved Feshbach–Villars charge, node counting, and numerical-domain convergence. The regularised Coulomb interaction is treated through full-line Loudon matching, which clarifies the cutoff dependence of the regular odd–even pairs and separates them from the isolated core branch. Symmetric power-exponential, Pöschl–Teller, and localised Woods–Saxon wells are analysed by parity-resolved inward shooting with bounded residual functions and explicit convergence tests. The pure-vector Cornell interaction is excluded because its large-distance solutions remain oscillatory, whereas a one-sided Woods–Saxon step is shown not to support a non-trivial square-integrable state on the full line under simultaneous decay conditions. A signed search also follows a Pöschl–Teller branch through zero energy and confirms a negative-energy continuation with positive integrated Feshbach–Villars charge. The comparison among the four models separates spatial localisation, asymptotic component mixing, and integrated component content, showing that these diagnostics need not follow the same trend. The resulting framework provides reproducible benchmarks for relativistic scalar bound-state calculations. Full article

In this study, we design a new magnetic field measurement model. Specifically, we use a single two-level atom coupled to two cavities to construct a parity–time-symmetric system supporting a third-order exceptional point. If a perturbation is applied to the atomic transition frequency, the eigenvalue of the system will change, and the eigenvalue change is proportional to the cubic root of the perturbation. If the perturbation comes from the magnetic field, a sensitive magnetic field measurement device is formed. By introducing gain and loss via the input–output field, we realize a third-order EP in the non-Hermitian Hamiltonian. Our analysis shows that the system exhibits a nonlinear response to magnetic field perturbations, leading to enhanced spectral sensitivity compared to conventional linear detectors. This equivalent EP based on a cavity-QED sensing scheme breaks the limitation of passive EP sensors and provides a new theoretical idea for the design of sensitive magnetic field measurement devices. Full article

Graphs $D(n,q)$ and their connected components $CD(n,q)$ were defined 30 years ago. We briefly review their applications to Extremal Graph Theory, Spectral Graph Theory, Algebraic Graph Theory, Symmetric Cryptography, and Theory of Low Density Parity Check Codes. We introduce several new algorithms of Noncommutative Cryptography based on these graphs of large girth. In particular we propose a modification of the Diffie–Hellman protocol in terms of the semigroup of walks of even length on the forest obtained as the projective limit of $D(n,q)$ and the homomorphic image of this monoid, acting on the vector space ${\left(F_q\right)}^n$ as the transformation group $G(n,q)$ of cubic polynomial transformations. The protocol allows users to compute a collision vector from ${\left(F_q\right)}^n$ in time $O\left(n^2\right)$. The security of these schemes rests on the complexity of the Conjugacy Power Problem for the affine Cremona semigroup of automorphisms of $F_q[x_1,x_2,\dots ,x_n]$. An inverse protocol of El Gamal type allows one to use this scheme for encryption or the creation of digital signatures. Several obfuscations of these algorithms are given. Full article

Surface-emitting distributed-feedback (SE-DFB) semiconductor lasers based on second-order gratings face a fundamental triple constraint: the spatial co-location of gain, grating feedback, and vertical radiation functions limits single-mode selectivity, surface extraction efficiency, and far-field beam quality simultaneously. We propose a quasi-parity-time (PT)-symmetric SE-DFB laser with separated gain and radiating-grating sections. In this design, the electrically injected gain section and the passive second-order grating section are placed in different regions along the cavity axis, thereby separating electrical injection from surface emission without epitaxial regrowth. Coupled-mode theory and two-dimensional finite-element simulations demonstrate that the resulting longitudinal non-Hermitian gain–loss asymmetry produces spatial-overlap-dependent threshold discrimination, enabling an isolated low-threshold lasing branch that remains separated from competing cavity modes over the investigated pump-parameter range. Under the HR–AR boundary condition, the proposed design achieves a threshold gain margin of $\mathsf{\Delta }g=12.4$${\mathrm{c}\mathrm{m}}^{-1}$, more than six times that of a conventional HR–AR DFB benchmark considered here, together with an upward surface extraction efficiency of 23.4% obtained from 2D FEM simulations. A simplified steady-state rate-equation estimate further suggests that the increased threshold margin can support strong side-mode suppression. The design imposes no regrowth requirement and is fully compatible with standard single-growth InP ridge-waveguide fabrication. Full article

Underwater wireless power transfer (UWPT) enables long-term deep-sea floor exploration by providing contactless energy replenishment for autonomous underwater vehicles (AUVs). However, conventional single-transmitter systems suffer reduced coupling and efficiency caused by high-loss underwater dielectrics and docking-induced perturbations. We propose a parallel-resonant dual-transmitter configuration based on the parity–time (PT) symmetric gain–loss-balanced modal framework. The proposed dual-transmitter single-receiver (DTSR) system forms a stronger and more symmetric field in the receiver than the single-transmitter baseline, counteracting the high-loss dielectric and improving the misalignment tolerance. According to the PT symmetry coupled-mode theory, we analyze how the quality factor and coupling strength determine the admissible PT-unbroken operating region over the docking-induced coupling range. An experimental prototype validates the analysis by comparing operating frequency and efficiency between DTSR and the single-transmitter baseline under distance (4.8–13.5 cm) and load (2.0–4.3 kΩ) variations. The results show that DTSR increases the critical coupling distance by 20–30% and reduces efficiency sensitivity to distance and load variations. These results suggest that the system can provide more robust and stable UWPT for AUV recharging under high-loss dielectric and perturbation, conducive to practically implementing AUV recharging in deep-sea operations. Full article

Defect modes in phononic crystals (PnCs) provide strongly localized resonances that are essential for frequency-dependent wave filtering and highly sensitive sensing. Their functionality increases greatly when their spectral characteristics can be externally tuned without altering the structural configuration. However, existing feedback control strategies rely on laminated piezoelectric defects, which have uniform electromechanical loading that causes voltage cancellation for even-symmetric defect modes. Consequently, only odd-symmetric defect bands can be manipulated effectively, which limits multi-band tunability. To overcome this constraint, we propose a segmented piezoelectric sensor–actuator design that enables symmetry-dependent feedback at the defect site. We develop a transfer-matrix analytical framework to incorporate complex-valued feedback gains directly into dispersion and transmission calculations. Analytical predictions demonstrate that real-valued feedback yields opposite stiffness modifications for odd- and even-symmetric modes. This enables the simultaneous tuning of both defect bands and induces an exceptional-point-like coalescence. In contrast, imaginary feedback preserves stiffness but modulates effective damping, generating a parity-dependent amplification-suppression response. The analytical results closely match those of fully coupled finite-element simulations, reducing computation time by more than two orders of magnitude. These findings demonstrate that segmentation-enabled feedback provides an efficient and scalable approach to tunable, multi-band, non-Hermitian wave control in piezoelectric PnCs. Full article

Conventional educational assessments enforce a rigid and symmetrical framework of identical question sequences upon a learner population inherently defined by asymmetry in cognitive capabilities and knowledge profiles. This mismatch results in inefficient measurement, where the uniform distribution of difficulty fails to mirror the heterogeneous nature of student learning. We address these topological and informational asymmetries through HARLA-ED, a hybrid framework combining deep knowledge modeling with intelligent question selection. The system integrates hierarchical cognitive graph networks to map the structural symmetries of concept dependencies while tracking evolving knowledge states across multiple time scales. By capturing both immediate working-memory constraints and long-term retention patterns, the model resolves the temporal asymmetry between learning and forgetting rates. A hierarchical reinforcement learning agent then orchestrates an assessment strategy through three decision levels: high-level planning determines diagnostic objectives, mid-level control sequences question types, and low-level actions select specific items. Crucially, the agent employs information-theoretic reward functions designed to restore distributional symmetry in assessment outcomes, ensuring demographic parity and minimizing algorithmic bias. Empirical results demonstrate a 47.5% average reduction in assessment duration compared to standard computer-adaptive tests while preserving measurement accuracy. The system successfully adapts to varying proficiency levels, effectively bridging the information asymmetry between the testing system and the learner’s true latent state. Full article

Optoelectronic oscillators (OEOs) have emerged as indispensable tools for generating low-phase-noise microwave and millimeter-wave signals, which are critical for a variety of high-performance applications. These include radar systems, satellite links, electronic warfare, and advanced instrumentation. The ability of OEOs to produce signals with exceptionally low phase noise makes them ideal for scenarios demanding high signal purity and stability. In radar systems, low-phase-noise signals enhance target detection accuracy and resolution, while, in communication networks, such signals enable higher data throughput and improved signal integrity over extended distances. Furthermore, OEOs play a pivotal role in precision instrumentation, where even minor noise can compromise the performance of sensitive equipment. This review examines the progress in OEO technology, transitioning from classical designs relying on long optical fiber delay lines to modern integrated systems that leverage photonic integration for compact, efficient, and tunable solutions. Key advancements, including classical setups, hybrid designs, and integrated configurations, are discussed, with a focus on their performance improvements in phase noise, side-mode suppression ratio (SMSR), and frequency tunability. A 20-GHz oscillation with an SMSR as high as 70 dB has been achieved using a classical dual-loop configuration. A 9.867-GHz frequency with a phase noise of −142.5 dBc/Hz @ 10 kHz offset has also been generated in a parity–time-symmetric OEO. Additionally, integrated OEOs based on silicon photonic microring resonators have achieved an ultra-wideband tunable frequency from 3 GHz to 42.5 GHz, with phase noise as low as −93 dBc/Hz at a 10 kHz offset. The challenges in achieving fully integrated OEOs, particularly concerning the stability and phase noise at higher frequencies, are also explored. This paper provides a comprehensive overview of the state of the art in OEO technology, highlighting future directions and potential applications. Full article

Integrated optical amplifiers are the building blocks of on-chip photonic systems, and they are often accompanied by a narrowband filter to limit noise. In this sense, a bandwidth-tunable optical amplifier with narrowband filtering function is crucial for on-chip optical circuits and radio frequency systems. The intrinsic loss and coupling coefficients between resonator and waveguide inherently limit the bandwidth. The parity-time symmetric coupled microresonators operating at exceptional points enable near zero bandwidth. In this study, we propose a parity-time symmetric coupled microresonators system operating near EPs to achieve a bandwidth of 46.4 MHz, significantly narrower than bandwidth of 600.0 MHz and 743.2 MHz achieved by two all-pass resonators with identical gain/loss coefficients. This system also functions as an optical bandwidth-tunable filter. The bandwidth tuning ranges from 175.7 MHz to 7.8 MHz as gain coefficient adjusts from 0.2 dB/cm to 0.4 dB/cm. Our scheme presents a unique method to obtain narrow bandwidth from two broadband resonators and serves as an optical bandwidth-tunable filter, thereby paving a new avenue for exploring non-Hermitian light manipulation in all-optical integrated devices. Full article

Parity-time (PT) symmetry has made encouraging progress in wireless power transmission (WPT), exhibiting significant advantages in terms of system robustness and transmission efficiency. However, there are still challenges that need to be addressed, particularly when classical schemes operate at a fixed frequency in the weak coupling region, where even minor changes in coupling strength can result in excessive current surges. This paper introduced a novel PT-symmetric WPT system featuring negative resistance constructed on the receiver side. We first established a theoretical framework for the classical two-coil PT-symmetric magnetically coupled resonant WPT system and subsequently extended it to incorporate the PT-symmetric WPT system with negative resistance on the receiver. This topological coil configuration facilitated stable power delivery over a broader range, with the capability of self-tuning frequency without requiring additional frequency modulation. This adaptability enabled the system to cater to diverse scenarios and opens up a novel avenue for practical applications of PT symmetry in WPT. Finally, we designed a 10 W prototype to demonstrate the effectiveness of our topology, and the experimental results aligned with our theoretical calculations, validating the feasibility and potential of our PT-symmetric WPT system. Full article

A capacitive wireless power transfer (CPT) system based on parity–time (PT) symmetry achieves constant output characteristics under distance variation without additionally increasing the system complexity of the control strategy, where the concept of PT symmetry is derived from quantum mechanics, and the systems satisfying PT symmetry are invariant under space and time inversion. However, the exact PT-symmetric region (i.e., strong coupling region) of the general system is limited by the symmetry of the structure and parameters. To overcome this limitation, a novel generalized parity–time (GPT)-symmetric CPT system is proposed in this article. According to the equivalent circuit method, the circuit model of the proposed system is built, and the transfer characteristics are analyzed. Furthermore, a prototype is implemented to verify the feasibility of the proposed CPT system. The results show that the PT-symmetric region is extended by 169.23% compared with the traditional PT-based CPT system, and a constant output power of 21.5 W is transferred with a constant transfer efficiency of 90%. Full article

When two resonators of coupled silicon resonators are identical and the gain on one side is equal to the loss on the other side, a parity-time (PT) symmetric-coupled silicon resonator is formed. As non-Hermitian systems, the PT-symmetric systems have exhibited many special properties and interesting phenomena. This paper proposes the strain-induced frequency splitting in PT symmetry-coupled silicon resonators. The frequency splitting of the PT system caused by strain perturbations is derived and simulated. Theory and simulation both indicate that the PT system is more sensitive to strain perturbation near the exceptional point (EP) point. Then, a feedback circuit is designed to achieve the negative damping required for PT symmetry. Based on a simple silicon-on-insulator (SOI) process, the silicon resonator chip is successfully fabricated. After that, the PT-symmetric-coupled silicon resonators are successfully constructed, and the frequency splitting phenomenon caused by strain is observed experimentally. Full article

With the rapid development in sensor network technology, the complexity and diversity of application scenarios have put forward more and more new requirements for inductor–capacitor (LC) sensors, for instance, multi-parameter simultaneous monitoring. Here, the parity–time (PT) symmetry concept in quantum mechanics is applied to LC passive wireless sensing. Two or even three parameters can be monitored simultaneously by observing the frequency response of the reflection coefficient at the end of the readout circuit. In particular, for three-parameter detection, a novel detection method is studied to extract the three resonant frequencies of the system through the phase–frequency characteristics of the reflection coefficient, which has never appeared in the previous literature on PT symmetry. The changes in three resonant frequencies are in response to changes in the three parameters in the environment. We show theoretically and demonstrate experimentally that the PT-symmetric LC sensor can realize multi-parameter measurement using a series LCR circuit as the sensor and a symmetric adjustable LCR circuit as the readout circuit. Our work paves the way for applying PT symmetry in multi-parameter detection. Full article

In this paper, a dual-wavelength narrow-linewidth fiber laser based on parity-time (PT) symmetry theory is proposed and experimentally demonstrated. The PT-symmetric filter system consists of two optical couplers (OCs), four polarization controllers (PCs), a polarization beam splitter (PBS), and cascaded fiber Bragg gratings (FBGs), enabling stable switchable dual-wavelength output and single longitudinal-mode (SLM) operation. The realization of single-frequency oscillation requires precise tuning of the PCs to match gain, loss, and coupling coefficients to ensure that the PT-broken phase occurs. During single-wavelength operation at 1548.71 nm (λ₁) over a 60-min period, power and wavelength fluctuations were observed to be 0.94 dB and 0.01 nm, respectively, while for the other wavelength at 1550.91 nm (λ₂), fluctuations were measured at 0.76 dB and 0.01 nm. The linewidths of each wavelength were 1.01 kHz and 0.89 kHz, with a relative intensity noise (RIN) lower than −117 dB/Hz. Under dual-wavelength operation, the maximum wavelength fluctuations for λ₁ and λ₂ were 0.03 nm and 0.01 nm, respectively, with maximum power fluctuations of 3.23 dB and 2.38 dB. The SLM laser source is suitable for applications in long-distance fiber-optic sensing and coherent LiDAR detection. Full article