Sahar H. Nazra

Introduction: Efficacy of conjugated photodynamic therapy (PDT)/chemotherapy in the treatment of breast cancer is summarized in this review. Alternative chemo drugs (CDs) and photosensitizers (PSs) are irradiated with lasers at various wavelengths to verify the toxicity of photo-stimulated compounds. Methods: The conducted studies on conjugated photodynamic and chemotherapy treatment of breast cancer across three phases (in vitro, in vivo and clinical) from 2014 to 2025 are summarized in this review. Articles on laser-activated therapies are evaluated, and irrelevant studies have been omitted. The records are sorted based on CD/PS typ laser irradiation wavelength, and energy. Results: Synergic PDT+chemotherapy is a promising method for breast cancer treatment according to in vitro, in vivo and clinical studies. Nevertheless, the clinical translation of the conjugated approach still meets unresolved issues, including long-term safety, limited selectivity, and biodistribution challenges. Conclusion: The hybrid treatment exhibits superior advantages in tumor shrinkage, high selectivity, inhibition of malignant cells, overcoming drug resistance, and metastasis control. Furthermore, the conjugate treatments mitigate the side effects of high-dosage chemotherapy.

Pseudo-Schouten symmetric spacetimes are examined under a differential condition on the Schouten tensor, producing explicit relations for the scalar curvature, the Ricci tensor, and the associated covector field. It is shown that the associated covector is a Ricci eigenvector and is irrotational, acceleration-free, and vorticity-free. Corresponding reductions of the Ricci tensor are obtained, including perfect fluid forms under suitable conditions. Ricci recurrent and Ricci semi-symmetric cases are also considered, leading to further curvature restrictions and compatibility properties involving the Riemann and Weyl tensors. In vacuum [Formula: see text]-gravity, the Ricci tensor is decomposed into a perfect fluid part and an anisotropic term governed by the shear tensor, equivalently by the electric part of the Weyl tensor. A comoving frame analysis shows that the shear vanishes, so that the associated covector is torse-forming and the spacetime reduces to a generalized Robertson–Walker model, and in four dimensions to a Robertson–Walker spacetime. Moreover, the Weyl tensor is divergence-free and annihilated by the associated covector. These results give a geometric characterization of pseudo-Schouten symmetric spacetimes in both the general setting and the vacuum [Formula: see text]-gravity case.

This paper investigates relativistic magneto-fluid spacetimes admitting generalized almost Schouten solitons and their gradient analogs within the framework of the Einstein field equations. By coupling the soliton structure with the magneto-fluid energy–momentum tensor, explicit relations are obtained between geometric invariants of spacetime and physical quantities such as fluid density, pressure, magnetic field strength, and magnetic permeability. Under quasi-conformal, [ ]-flat, and [ ]-flat curvature conditions, strong rigidity results are derived, showing that the spacetime reduces to an Einstein or constant-curvature model under natural algebraic assumptions. The influence of soliton parameters on the expanding, steady, and shrinking behavior of the spacetime is characterized explicitly. Energy conditions associated with generalized almost Schouten solitons are analyzed, and it is shown that the timelike convergence condition implies both the strong and null convergence conditions, allowing the application of Penrose-type singularity theorems and guaranteeing the existence of trapped surfaces and black hole regions. The harmonic and Schrödinger–Ricci harmonic properties of the soliton-induced [ ]-form are completely characterized, while in the gradient case the soliton potential satisfies a Poisson-type equation governing its qualitative behavior. Several physically relevant regimes, including dust, dark matter, radiation, and vanishing magnetic permeability, are examined, and an explicit four-dimensional Lorentzian example is constructed to confirm the existence of nontrivial generalized almost Schouten solitons.

In this study, we consider the problem of how to characterize a surface family from a given common asymptotic curve in a three-dimensional Galilean space [ ]. We obtain a parametrization of the surface such that the surface tangent plane is coincident with the curve osculating plane. The extension to ruled surfaces is also outlined. We illustrate this method by presenting some examples.

In this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4. Using the Serret–Frenet frame of the given geodesic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be geodesic. We illustrate this method by presenting some examples.

Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained.

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.