Search Results (84)

We investigate gradient Einstein solitons admitting closed conformal vector fields. Under constant scalar curvature, we establish rigidity results showing that complete solitons with non-parallel closed conformal vector fields are isometric to Euclidean space. In the non-compact case, we prove that solitons with homothetic closed conformal vector fields are locally conformally flat in dimension four and possess a harmonic Weyl tensor in higher dimensions. Furthermore, we obtain a classification result, showing that such solitons admit a warped product structure with a one-dimensional base and space form fibers. Full article

The present paper is aimed at studying the convergence of the Yamabe flow in the case of noncompact solitons. The more specified example of locally conformally flat noncompact solitons is addressed with the aim to newly analyse the qualities of the Ricci scalar. The particular case of noncompact pseudo-Riemannian solitons is studied; moreover, in the instances of Schwarzschild and Generalized-Schwarzschild geometries, rescalings of spherically symmetric weights are performed. For this purpose, new results are achieved as far as the considered structures are concerned. The Myers Theorem is upgraded as the new Myers paradigm of spacetime-dimensional manifolds, where the Einstein Field Equations can now be taken into account. In particular, the Myers Theorems are studied here as far as their new implementation in General Relativity Theory is concerned. As a first important result, the Myers mean curvature is found to coincide with the Ricci scalar in General Relativity Theory, where the 4-position of the observer, from which the 4-velocity 4-vector is calculated from, is taken as that of the observer solidal with the reference frame of the photon. The following results are also of relevance. In more detail, the umbilicity conditions are applied. At a further step, the role of the umbilicity conditions in GR after the Myers Theorems are studied for weighted manifolds and specific new implications of weighted manifolds are developed. The description of the weighted Schwarzschild manifolds and that of the weighted Generalized-Schwarzschild manifolds are newly studied as follows: as a new finding, the Birkhoff Theorem is newly reconciled with the rotational ansatz of the metrised solitons, and the comparison with the previous results about the Brendle non-metrised solitons is accomplished with the outcome stressing the new roles of the new rescalings of the metric tensor with respect to the previous known results of the scaling of the metric tensor of the non-metrised solitons. In the present framework, these procedures allow one to prove the reconciliation of the EFEs with the Yamabe flow. The flow on the tipping lightcones is newly written. The umbilicity condition is studied in General Relativity after the upgrade of the Myers Theorems as far as the sectional curvatures are concerned; as a result, the Calabi–Bernstein description is implemented in General Relativity, as well as the Chen–Yau requirements, and the cases of weighted manifolds are taken into account. More specifically, the equal-time 2-dimensional space surfaces are studied analytically, onto which the weighted General-Relativistic solitons which satisfy the Einstein field equations after the Yamabe flow are projected due to the rotational ansatz. As an accessory introductory result, the class of Wu non-metrised solitons are proven to be discarded in several aspects of the Wu description as the conditions provided after the work of Wu are not compatible with metrisation. Full article

This manuscript presents a comprehensive taxonomy of Riemann solitons within the framework of a spacetime manifold endowed with a metric exhibiting both spatial homogeneity and rotational characteristics. Furthermore, we undertake an analysis to determine the geometric nature of these solitons by establishing their correspondence to Killing vector fields, Ricci collineation vector fields, and gradient vector fields. Full article

In this paper, we examine Ricci–Bourguignon solitons on locally decomposable golden Riemannian manifolds of constant golden sectional curvature. First, we establish an explicit expression for the soliton constant in terms of the golden structure and the Bourguignon parameter. Second, we explore the geometry of these solitons when the potential vector fields are Killing, conformal Killing, homothetic, or concurrent. Finally, we initiate the study of golden Ricci–Bourguignon solitons, determine their soliton constants, and examine their properties under the specific potential vector fields. Full article

In this investigation, we examine the geometric character of almost Riemann solitons and gradient almost Riemann solitons in the context of perfect fluid solutions of the Einstein equations that admit a torse-forming vector field $\zeta$. We first examine the conditions on the scalar curvature, which are necessary for the existence of an almost Riemann soliton or a gradient almost Riemann soliton in such solutions. We then examine the case of several physically reasonable types of perfect fluids, such as dark fluids, dust-filled universes, and the radiation-dominated epoch. We also show that any spacetime bearing an almost Riemann soliton with a conformal potential vector field must necessarily have an Einstein geometry. In addition, in the case of a perfect fluid spacetime with a torse-forming vector field, given the fulfillment of the almost Riemann soliton compatibility equation and $Q·P=0$, the scalar curvature of the spacetime must be constant. Finally, a rigidity theorem states that any parallel symmetric $(0,2)$-tensor defined on the spacetime must be a constant multiple of the metric tensor. Full article

This paper studies the conformal geometry of complete gradient Schouten solitons (GSSs) admitting closed conformal vector fields (CVFs). We establish rigidity and characterization results for nonparallel, homothetic closed CVFs under the assumption that the gradient of the scalar curvature is parallel to the CVF. It is shown that such manifolds are isometric to Euclidean space. Moreover, complete noncompact GSSs with constant scalar curvature are locally conformally flat in dimension four and have harmonic Weyl curvature in higher dimensions. Finally, we prove that these manifolds are totally umbilical if and only if their scalar curvature is constant, and they form warped products with space forms. 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

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

Computational and machine learning approaches play a pivotal role in identifying, characterizing, and targeting noncanonical DNA structures, including G-quadruplexes, Z-DNA, hairpins, and triplexes. These configurations play critical roles in maintaining genomic stability, facilitating DNA repair, and regulating chromatin organization. Although the human genome predominantly adopts the B DNA conformation, evidence indicates that non-B DNA forms exert significant influence on gene expression and disease development. This highlights the need for dedicated computational frameworks to systematically investigate these alternative structures. Machine learning model, encompassing supervised and unsupervised algorithms such as K Nearest Neighbors, Support Vector Machines, and deep learning architectures including Convolutional Neural Networks, have shown considerable potential in predicting sequence motifs predisposed to forming non-B DNA conformations. These predictive tools contribute to identifying genomic regions associated with disease susceptibility. Complementary bioinformatics platforms and molecular docking tools, notably Auto Dock, along with chemical libraries like ZINC, facilitate the virtual screening of small molecules targeting specific DNA structures. Stabilizers of G quadruplexes, exemplified by CX 5461, have demonstrated therapeutic promise in BRCA-deficient cancers, highlighting the translational impact of computational methods on drug discovery. Anticipating DNA structural shifts opens new avenues in personalized medicine for complex diseases, with computational chemistry and machine learning deepening our understanding of DNA topology and guiding smarter ligand design. The integrated approach proposed in this review addresses the previous studies performed in this field and highlights the current limitations in structural genomics and advances the development of precision therapeutics aligned with individual genomic profiles. Full article

Compaction quality control in earthworks and pavements still relies mainly on density-based acceptance referenced to laboratory Proctor tests, which are costly, time-consuming, and spatially sparse. Lightweight dynamic cone penetrometer (LDCP) provides rapid indices, such as $q_{d0}$ and $q_{d1}$, yet acceptance thresholds commonly depend on ad hoc, site-specific calibrations. This study develops and validates a supervised machine learning framework that estimates $q_{d0}$, $q_{d1}$, and $Z_c$ directly from readily available soil descriptors (gradation, plasticity/activity, moisture/state variables, and GTR class) using a multi-campaign dataset of $n=360$ observations. While the framework does not remove the need for the standard soil characterization performed during design (e.g., W, ${\gamma }_{d,\mathrm{field}}$, and $R{C}_{\mathrm{SPC}}$), it reduces reliance on additional LDCP calibration campaigns to obtain device-specific reference curves. Models compared under a unified pipeline include regularized linear baselines, support vector regression, Random Forest, XGBoost, and a compact multilayer perceptron (MLP). The evaluation used a fixed 80/20 train–test split with 5-fold cross-validation on the training set and multiple error metrics ($R^2$, RMSE, MAE, and MAPE). Interpretability combined SHAP with permutation importance, 1D partial dependence (PDP), and accumulated local effects (ALE); calibration diagnostics and split-conformal prediction intervals connected the predictions to QA/QC decisions. A naïve GTR-average baseline was added for reference. Computation was lightweight. On the test set, the MLP attained the best accuracy for $q_{d1}$ ($R^2=0.794$, RMSE $=5.866$), with XGBoost close behind ($R^2=0.773$, RMSE $=6.155$). Paired bootstrap contrasts with Holm correction indicated that the MLP–XGBoost difference was not statistically significant. Explanations consistently highlighted density- and moisture-related variables (${\gamma }_{d,\mathrm{field}}$, $R{C}_{\mathrm{SPC}}$, and W) as dominant, with gradation/plasticity contributing second-order adjustments; these attributions are model-based and associational rather than causal. The results support interpretable, computationally efficient surrogates of LDCP indices that can complement density-based acceptance and enable risk-aware QA/QC via conformal prediction intervals. Full article

The nature of the $F(R,T)$-gravity in conjunction with the quark matter fluid $\left(QMF\right)$ is examined in this research note. In the $F(R,T)$-gravity framework, we derive the equation of state for the $QMF$ in the form of: $F(R,T)=F_1\left(R\right)+F_2\left(T\right)$ and the model of $F\left(R\right)$-gravity. We also discuss how the quark matter supports the Ricci solitons with a conformal vector field in $F(R,T)$-gravity. In this continuing work, we give estimates for the pressure and quark density in the phantom barrier period and the radiation epoch, respectively. Additionally, we use Ricci solitons to identify several black hole prospects and energy requirements for quark matter fluid spacetime ($QMF$-spacetime) connected with $F(R,T)$-gravity. Furthermore, in the $F(R,T)$-gravity model connected with $QMF$, we also discuss some applications of the Penrose singularity theorem in terms of Ricci solitons with a conformal vector field. Finally, we deduce the Schrödinger Equation using the equation of state of the $F(R,T)$-gravity model connected with $QMF$, and we uncover some constraints that imply the existence of compact quark stars of the $Ia$-supernova type in the $QMF$-spacetime with $F(R,T)$-gravity. Full article

In the present research note, we explore the nature of the conformal Ricci solitons on the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ that is a modification of general relativity. Furthermore, we deal with a subcase of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity model coupled with a perfect fluid, which admits conformal Ricci solitons with a time-like concircular vector field. Using the steady conformal Ricci soliton, we derive the equation of state for the perfect fluid in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model. In this series, we convey an indication of the pressure and density in the phantom barrier period and the stiff matter era, respectively. Finally, using a conformal Ricci soliton with a concircular vector field, we study the various energy constraints, black holes, and singularity circumstances for a perfect fluid coupled to $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity. Lastly, employing conformal Ricci solitons, we formulate the first law of thermodynamics, enthalpy, and the particle production rate in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity and orthodox gravity. Full article

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is invertible, we derive the algebraic soliton equations for left-invariant metrics and classify all left-invariant generalized cross-curvature solitons (for the generalized equation ${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg$) on the standard 3D Lorentzian Lie algebra types (unimodular Types Ia, Ib, II, and III and non-unimodular Types IV.1, IV.2, and IV.3). For each Lie algebra type, we state the necessary and sufficient algebraic conditions on the structure constants, provide explicit formulas for the soliton vector fields X (when they exist), and compute the soliton parameter $\lambda$ in terms of the structure constants and the parameter $\rho$. Our results include several existence families, explicit nonexistence results (notably for Type Ib and Type IV.3), and consequences linking the existence of left-invariant solitons with local conformal flatness in certain cases. The classification yields new explicit homogeneous generalized cross-curvature solitons in the Lorentzian setting and clarifies how the parameter $\rho$ modifies the algebraic constraints. Examples and brief geometric remarks are provided. Full article

Through the years, the asymmetry in the constitutive relations that define the electric and magnetic polarization, P and M, respectively, by the relevant vector field, E and H, has been imprinted, rather arbitrarily, in Maxwell’s equations. Accordingly, in linear, homogeneous, and isotropic (LHI) materials, the electric and magnetic polarization are defined via P = χ_eε₀E (‘P-E, χ_e’ formulation; 0 ≤ χ_e < ∞) and M = χ_mH (‘M-H, χ_m’ formulation; −1 ≤ χ_m < ∞), respectively. Recently, the constitutive relation of the polarization was revisited in LHI dielectrics by introducing an electric susceptibility, χ_ε, which couples linearly the reverse polarization, $\tilde{\mathbf{P}}$ = −P, with the electric displacement D through $\tilde{\mathbf{P}}$ = $\mathsf{\chi }$_εD (‘P-D, χ_ε’ formulation; −1 ≤ χ_ε ≤ 0). Here, the ‘P-D, χ_ε’ formulation is generalized for the time-dependent case. It is documented that the susceptibility and polarization of LHI dielectric and magnetic materials can be described by the ‘P-D, χ_ε’ and ‘M-H, χ_m’ formulation, respectively, on a common basis. To this end, the depolarizing effect is taken into account, which unavoidably emerges in realistic specimens of limited size, by introducing a series scheme to describe the evolution of polarization and calculate the extrinsic susceptibility. The engagement of the depolarizing factor N (0 ≤ N≤ 1) with the accompanying convergence conditions dictates that the intrinsic susceptibility of LHI materials, whether electric or magnetic, should range within [−1, 1]. The ‘P-D, χ_ε’ and ‘M-H, χ_m’ formulations conform with this expectation, while the ‘P-E, χ_e’ does not. Remarkably, Maxwell’s equations are unaltered by the ‘P-D, χ_ε’ formulation. Thus, all time-dependent processes of electromagnetism described by the standard ‘P-E, χ_e’ approach, are reproduced equivalently, or even advantageously, by the alternative ‘P-D, χ_ε’ formulation. 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