Search Results (248)

We investigated an exact solution in a conformal invariant Randall-Sundrum 5D warped brane world model on a time dependent Kerr-like spacetime. The singular points are determined by a quintic polynomial in the complex plane and fulfills Cauchy’s theorem on holomorphic functions. The solution, which is determined by a first-degree differential equation, shows many similarities with an instanton. In order to describe the quantum mechanical aspects of the black hole solution, we apply the antipodal boundary condition. The solution is invariant under time reversal and also valid in Riemannian space. Moreover, CPT invariance in maintained. The vacuum instanton solution follows from the 5D as well as the effective 4D brane equations, only when we allow the contribution of the projected 5D Weyl tensor on the brane (the KK-‘particles’). The topology of the effective 4D space of the brane is the projective $\mathbb{R}{P}^3$ (elliptic space) by identifying antipodal points on $S^3$. The 5D is completed by applying the Klein bottle embedding and the ${\mathbb{Z}}_2$ symmetry of the RS model. This model fits very well with the description of the Hawking radiation, which remains pure. We have also indicated a possible way to include fermions. Our 5D space admits a double cover of $S^3$ and after fibering to the $S^2$, we obtain the effective black hole horizon. The connection with the icosahedron discrete symmetry group is investigated. It seem that Bekenstein’s conjecture that the area of a black hole is quantized, could be applied to our model. Full article

This paper investigates pseudo-symmetric space–times within two interrelated frameworks: vacuum $f\left(R\right)$-gravity and Gray’s seven canonical decomposition subspaces. First, it is established that any conformally flat pseudo-symmetric space–time satisfying the vacuum field equations of $f\left(R\right)$-gravity necessarily corresponds to a perfect fluid. Subsequently, a detailed analysis of Gray’s subspaces reveals the following structural results: In the trivial and $\mathsf{𝒜}$ subspaces, pseudo-symmetric space–times are Ricci-simple and Weyl-harmonic, and thus are necessarily generalized Robertson–Walker space–times. In the $\mathcal{B}$ and $\mathsf{𝒜}\oplus \mathcal{B}$ subspaces, the associated time-like vector field ${\xi }^l$ is shown to be an eigenvector of the Ricci tensor with the eigenvalue $-R/2$. Furthermore, for a perfect fluid pseudo-symmetric space–time obeying $f\left(R\right)$-gravity and belonging to the trivial, $\mathsf{𝒜}$, $\mathcal{B}$, or $\mathsf{𝒜}\oplus \mathcal{B}$ subspaces, the isotropic pressure p and energy density $\sigma$ are proven to be constants. Additionally, it is demonstrated that Gray’s $\mathcal{I}$ subspace reduces to the $\mathcal{B}$ subspace in the pseudo-symmetric setting. Finally, under specific geometric conditions, pseudo-symmetric space–times in the $\mathcal{I}\oplus \mathsf{𝒜}$ and $\mathcal{I}\oplus \mathcal{B}$ subspaces are also shown to admit perfect fluid representations. These results collectively clarify the geometric and physical constraints imposed by pseudo-symmetry within $f\left(R\right)$-gravity and Gray’s classification scheme. Full article

Symmetry and asymmetry are foundational design principles in artificial intelligence, defining the balance between invariance and adaptability in multimodal learning systems. In audio-visual speaker verification, where speech and lip-motion features are jointly modeled to determine whether two utterances belong to the same individual, these principles govern both fairness and discriminative power. In this work, we analyze how symmetry and asymmetry emerge within a gated-fusion architecture that integrates Time-Delay Neural Networks and Bidirectional Long Short-Term Memory encoders for speech, ResNet-based visual lip encoders, and a shared Conformer-based temporal backbone. Structural symmetry is preserved through weight-sharing across paired utterances and symmetric cosine-based scoring, ensuring verification consistency regardless of input order. In contrast, asymmetry is intentionally introduced through modality-dependent temporal encoding, multi-head attention pooling, and a learnable gating mechanism that dynamically re-weights the contribution of audio and visual streams at each timestep. This controlled asymmetry allows the model to rely on visual cues when speech is noisy, and conversely on speech when lip visibility is degraded, yielding adaptive robustness under cross-modal degradation. Experimental results demonstrate that combining symmetric embedding space design with adaptive asymmetric fusion significantly improves generalization, reducing Equal Error Rate (EER) to 3.419% on VoxCeleb-2 test dataset without sacrificing interpretability. The findings show that symmetry ensures stable and fair decision-making, while learnable asymmetry enables modality awareness together forming a principled foundation for next-generation audio-visual speaker verification systems. Full article

In this research paper, we introduce the notions of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons. We study the hyperbolic ∗-Ricci solitons on a three-dimensional $\epsilon$-trans-Sasakian manifold. Specifically, we determine the hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold with a conformal vector field and a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-type vector field. Using hyperbolic ∗-Ricci solitons with a conformal vector field, we discuss some geometric symmetries on a three-dimensional ($\epsilon$)-trans-Sasakian. In addition, we exhibit the nature of gradient hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold endowed with a scalar concircular field. Moreover, we demonstrate an example on a three-dimensional ($\epsilon$)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons and find the rate of change of the hyperbolic ∗-Ricci solitons within the same example. Lastly, we also introduce the concept of modified second hyperbolic ∗-Ricci solitons. Full article

This study investigates the conformable nonlinear Schrödinger equation (NLSE) with self-phase modulation (SPM) and Kudryashov’s generalized refractive index, crucial for pulse propagation in optical fibers. By applying the modified simplest equation method, we derive several novel soliton solutions and investigate their dynamic behavior within the NLSE framework enhanced with a conformable derivative. The governing conformable NLSE also exhibits symmetry patterns that support the structure and stability of the constructed soliton solutions, linking this work directly with symmetry-based analysis in nonlinear wave models. Furthermore, various graphs are presented through 2D, 3D, and contour plots. These visualizations highlight different soliton profiles, including kink-type, wave, dark, and bell-shaped solitons, showcasing the diverse dynamics achievable under this model, influenced by SPM and Kudryashov’s generalized refractive index. The influence of the conformable parameter and temporal effects on these solitons is also explored. These findings advance the understanding of nonlinear wave propagation and have critical implications for optical fiber communications, where managing pulse distortion and maintaining signal integrity are vital. Full article

(1) Background: Non-invasive instrumental measurement of swallowing acoustic signals has rested upon the assumptions of signal symmetry and reproducibility along the cervical region and has hence taken the form of single-point acquisition on optimal sites. This study aimed to (i) revisit such assumptions by adopting a conformal array of accelerometers, and hence (ii) lay the foundation for the future design of swallowing accelerometry. (2) Methods: Thirteen young healthy individuals, including eight females (mean age ± SD = 24.38 ± 0.92) and five males (mean age ± SD = 24 ± 3.74), were recruited. Swallowing mechano-acoustic signals of repeated swallowing trials were captured using conformal swallowing accelerometry. The peak intensities and frequencies as well as their respective peak times were extracted from six symmetrical and vertically aligned sites. (3) Results: Three-way ANOVAs with repeated measures suggested differences across trials and channels for both peak intensity and frequency. The additional interaction of bolus volume and repeated trials with a small effect size was also indicated in peak frequency. Intra-personal variability was indicated by coefficients of variance of the peak intensity and frequency of higher than 20%, with values varying within the 95% limits of agreement of at least 10 m/s² and 100 Hz, respectively. However, intra-trial comparisons of contra-lateral peak intensity and frequency also revealed a high degree of variability, with the 95% limits of agreement up to 12 m/s² and 240 Hz, respectively. On the other hand, the time points of intra-trial peak intensity and frequency showed a high degree agreement, suggesting the possibility of signal asymmetry. (4) Conclusions: The current findings not only confirmed the previous proposal of intra-personal variability but also demonstrated preliminary counterevidence to the longstanding assumption of signal symmetry. Alternatively, the use of conformal swallowing accelerometry is a promising option for the future design and implementation of non-invasive swallowing mechano-acoustic measurements. Full article

In this work, we develop a unified framework for Conformal Gravity and Noncommutative (Fuzzy) Gravity incorporating internal interactions. Our approach relies on two fundamental observations: first, the dimensions of a curved manifold and those of its tangent group need not coincide, and second, both gravitational models can be formulated as gauge theories. We begin with a discussion of the gauge-theoretic formulation of gravitational dynamics, emphasizing the role of diffeomorphism invariance. We then outline the constructions of Conformal Gravity and Fuzzy Gravity within this formalism. Building on an extension of the four-dimensional tangent group, we propose a scheme that unifies the two theories while naturally incorporating internal degrees of freedom. We further investigate the low-energy limits that emerge after appropriate spontaneous symmetry-breaking mechanisms, and we comment on potential observational signatures—particularly those associated with cosmic strings and their imprint on gravitational-wave spectra. Full article

Correct conformation and symmetry of bilateral hooves are associated with a properly balanced hoof, which has a significant impact on the biomechanics of movement and reduces the risk of injury. The aim of the study was to compare solear aspect dimensions of bilateral hooves in forelimbs (hoof width, length, frog width, length, medial and lateral diagonals) with regard to sex, breed and age of horses. The dimensions were measured with a calliper in 100 horses of four breeds. The results showed that the breed factor significantly affected all dimensions. Age was significant for some variables, while sex was not a significant factor. The hoof solear dimensions did not differ between mares and geldings, decreased with a corresponding decrease in horse breed size and often increased with age. Significant differences in the mean dimensions between bilateral hooves were rare and did not show a clear tendency across the population. However, considering the prevalence, almost 1/3 of horses showed asymmetries in particular dimensions. It was concluded that there are no consistent, one-directional asymmetries in the solear aspect of bilateral hooves in horse forelimbs despite frequent two-directional asymmetries occurring in horses. Trimmers should regard the findings to try to maintain hoof symmetry. Full article

This report focuses on the oxalate anion radical (C₂O₄^●−) formed during γ-radiolysis of polycrystalline oxalates: protonated oxalic acid (H₂C₂O₄·2H₂O), caesium oxalate (Cs₂C₂O₄), and potassium oxalate monohydrate (K₂C₂O₄·H₂O). Irradiation at 77 K generates stable radical species, revealed by EPR spectroscopy and supported by DFT calculations. In H₂C₂O₄·2H₂O, the primary axial signal (g_avg = 2.0035) is shown to arise from the structural relaxation of the HC₂O₄∙ radical into the intrinsically stable non-planar (D_2d) conformation, resolving the symmetry conflict with the planar crystal precursor. Numerical deconvolution confirmed the co-existence of this radical with the secondary HCO₂^● species, exhibiting distinct relaxation characteristics. In Cs₂C₂O₄, the broad isotropic signal (g ≈ 2.008) is attributed to the D_2d form. Quantitative analysis proved a sharp, thermodynamically driven structural conversion (D_2d→D_2h) upon annealing above 220 K, where the D₂h conformer (g_avg ≈ 2.011) becomes the dominant species (≈73%). In K₂C₂O₄·H₂O, the C₂O₄^●− radical undergoes rapid decomposition into the CO₂^●− radical (g_avg ≈ 2.0007) due to the kinetic instability of the primary species in that matrix. Our findings underscore the crucial role of computational assistance and quantitative numerical fitting in EPR studies: DFT provided crucial assistance and yielded satisfactory agreement in most cases, while clarifying the structural and kinetic stability governed by the local cationic environment. The stability of the most resistant radical forms persists up to 430 K in the caesium salt. Full article

We investigate the geometric and physical properties of the h-almost conformal $\eta$-Ricci–Bourguignon soliton and its gradient form by employing a barotropic equation of state within the framework of $f(\mathcal{R},\mathcal{T})$ gravity. We derive this barotropic equation of state under the assumption that the matter content of $f(\mathcal{R},\mathcal{T})$ gravity is modeled by a barotropic perfect fluid. We also examine the way in which these soliton structures both reveal and limit the underlying symmetries of the spacetime geometry. Furthermore, we obtain modified Poisson and Liouville equations associated with these solitons in such a gravitational setting. Additionally, we explore certain harmonic aspects of the h-almost conformal $\eta$-Ricci–Bourguignon soliton on a spacetime filled with a barotropic perfect fluid, considering a harmonic potential function $\mathrm{\Psi }$. Finally, we present physical interpretations of the conformal pressure $\tilde{p}$ in the context of the h-almost conformal $\eta$-Ricci–Bourguignon soliton within $f(\mathcal{R},\mathcal{T})$ gravity. Full article

Achieving a best-fit prosthetic socket is essential to comfort, functional performance, and long-term residual limb health in lower-limb amputees. To our knowledge, no previous study has quantitatively compared in vivo residual limb geometry under vacuum versus non-vacuum conditions using high-resolution computed tomography (CT). In this patient-specific case study of a bilateral knee-disarticulation (KD) amputee, both residual limbs were scanned under standardized conditions: one enclosed in a vacuum-compressed sleeve and the contralateral limb untreated as a natural control, thereby minimizing inter-subject variability. CT-based 3D reconstructions enabled volumetric and cross-sectional quantification, including symmetry/asymmetry analysis of paired limbs, while finite element analysis (FEA) assessed the biomechanical consequences for socket performance. Vacuum application resulted in a 4.1% reduction in total limb volume and a 5.3% reduction in mid-thigh cross-sectional area, with regionally asymmetric displacement of soft tissues. FEA demonstrated that vacuum-induced geometry reduced peak Von Mises stresses (27.43 MPa to 15.83 MPa), minimized maximum displacement (1.72 mm to 0.88 mm), and improved minimum factor of safety (~2.0 to ~3.0), while homogenizing contact pressure distribution (peak fell from 2.42 to 1.28 N/mm²). These findings provide preliminary CT-based evidence that vacuum application induces measurable morphological adaptations with implications for socket conformity, comfort, and load transfer. While limited to a single patient, this study highlights the potential of vacuum-induced modeling to inform personalized prosthetic socket design. Full article

This work presents new results concerning weighted Hardy-type inequalities within the framework of delta conformable fractional integrals on arbitrary time scales. The proposed approach unifies the treatment of inequalities across continuous and discrete domains, enabling the derivation of original forms in both settings. The obtained results exhibit symmetry with classical inequalities, and several integral and discrete inequalities arise as special cases. These findings extend and generalize known results and enrich the theory of integral inequalities in fractional and dynamic calculus, providing a versatile platform for further developments in symmetric and weighted inequality analysis. Full article

In this work, we introduce the conformabletriple Laplace–Sumudu transform (CTLST), a novel integral transform designed to solve both linear and nonlinear conformable FPDEs. This new approach builds on the recent development of the triple Laplace–Sumudu transform and incorporates the conformable derivative to extend its applicability to fractional models. We first present the foundational definitions and key properties of the CTLST, followed by its application to a variety of two- and three-dimensional conformable FPDEs. The effectiveness of the proposed method is demonstrated through several examples, where exact and approximate solutions are derived, illustrative 3D plots are presented, and symmetry analysis is employed to verify the obtained results. The CTLST provides a promising analytical tool for tackling complex conformable FPDEs in mathematical physics and engineering. Full article

In this paper, we investigate the geometric realization of $\mathrm{Spin}\left(8\right)$ triality through vector fields on the octonionic algebra $\mathbb{O}$. The triality automorphism group of $\mathrm{Spin}\left(8\right)$, isomorphic to ${\mathfrak{S}}_3$, cyclically permutes the three inequivalent 8-dimensional representations: the vector representation V and the spinor representations $S_{+}$ and $S_{-}$. While triality automorphisms are well known through representation theory, their concrete geometric realization as flows on octonionic space has remained unexplored. We construct explicit smooth vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ on $\mathbb{O}\cong {\mathbb{R}}^8$ whose flows generate infinitesimal triality transformations. These vector fields have a linear structure arising from skew-symmetric matrices that implement simultaneous rotations in three orthogonal coordinate planes, providing the first elementary geometric description of triality symmetry. The main results establish that these vector fields preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$, demonstrating fundamental compatibility between geometric flows and octonionic algebra. We prove that the standard Euclidean metric on $\mathbb{O}$ is triality-invariant and classify all triality-invariant Riemannian metrics as conformal to the Euclidean metric with a conformal factor depending only on the isotonic norm. This classification employs Schur’s lemma applied to the irreducible $\mathrm{Spin}\left(8\right)$ action, revealing the rigidity imposed by triality symmetry. We provide a complete classification of triality-symmetric minimal surfaces, showing they are locally isometric to totally geodesic planes, surfaces of revolution about triality-fixed axes, or surfaces generated by triality orbits of geodesic curves. This trichotomy reflects the threefold triality symmetry and establishes correspondence between representation-theoretic decomposition and geometric surface types. For complete surfaces with finite total curvature, we establish global classification and develop explicit Weierstrass-type representations adapted to triality symmetry. Full article

This work establishes the first derivation of geometry-dependent Kirchhoff’s laws via conformal symmetry, enabling new types of self-sustaining circuits unattainable in classical lumped-element theory. Building on Bessel-Hagen’s extension of Noether’s theorem to Maxwell’s equations, we develop a conformal circuit formalism that fundamentally extends traditional circuit theory through two key innovations: (1) Geometry-dependent weighting factors ($w_i\propto a_i^{-1}$) in Kirchhoff’s laws derived from scaling symmetry; (2) A dilaton-like field ($\delta$) mediating energy exchange between circuits and conformal backgrounds. Unlike prior symmetry applications in electromagnetism, our approach directly maps the 15-parameter conformal group to component-level circuit transformations, predicting experimentally verifiable phenomena: (i) $10.2\%$ deviations from classical current division in RF splitters; (ii) $4.2\%$ resonant frequency shifts with $2.67\times$ Q-factor enhancement; (iii) Power-law scaling ($J_z\propto a^{-2}$) in cylindrical conductors. This theoretical framework proposes how conformal symmetry could enable novel circuit behaviors, including potential self-sustaining oscillations, subject to experimental validation. Full article