Narrow Results

According to the general theory of relativity, a massive body induces curvature in the surrounding space-time. In this study, the surface curvature (SC) of neutron stars is computed using various curvature quantities derived from the relativistic mean-field (RMF), density-dependent RMF (DDRMF) and Skyrme–Hartree–Fock equations of states. Neutron star properties, including mass, radius, compactness and central density, are calculated utilizing the Tolman–Oppenheimer–Volkoff equations. The analysis reveals a significant cubic correlation between the SC and compactness for the canonical 1.4$M_{\odot }$ neutron star, with a correlation coefficient of 0.99, indicating an almost linear relationship. A similarly significant inverse cubic correlation is observed between the SC and the radius of the canonical star. However, these correlations diminish for the maximum mass NS. Furthermore, a universal relation between the SC and the dimensionless tidal deformability ($\mathrm{\Lambda }$) for the canonical neutron star is established. Using the tidal deformability constraint of GW170817 (${\mathrm{\Lambda }}_{1.4}=190_{-120}^{+390}$), the surface curvature is limited to SC${}_{1.4}(10^{14})=2.87_{-0.78}^{+0.30}$ at a confidence level of 90%. Furthermore, the tidal deformability constraint of the secondary component in the GW190814 event (${\mathrm{\Lambda }}_{1.4}=616_{-158}^{+273}$)¹ offers a more stringent limit, with the result of SC${}_{1.4}(10^{14})=2.03_{-0.36}^{+0.27}$. These findings indicate that the GW190814 event imposes more rigorous constraints on SC compared to GW170817.

We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan’s structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü–Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed.

There are significant identification errors in the vibration displacement identification of deep-water pile foundation bridge piers on soft soil foundations. Therefore, this paper proposes a new method for identifying the vibration displacement of deep-water pile foundation bridge piers on soft soil foundations based on multi-source information fusion. The characteristics of soft soil structure are analyzed and the stress relationship between soft soil and solid soil skeleton is calculated. Deep water pile foundation bridge piers on soft soil foundation determine the structural characteristics of deep water pile foundation bridge piers on soft soil foundation, construct displacement curves based on the elastic deformation curvature of deep water pile foundation bridge piers on soft soil foundation, and determine the curvature of each point when the bridge pier displacement occurs. Analyze the factors that cause the vibration of deep-water pile foundation bridge piers on soft soil foundations, and determine the characteristics of the vibration of deep-water pile foundation bridge piers on soft soil foundations by calculating the amplitude under different factors. Analyze the different failure modes and shear strength, bending strength, displacement ductility coefficient under the vibration conditions of bridge piers, and analyze the load on the bridge piers to complete the extraction of multi-source information on the vibration displacement of deep-water pile foundation bridge piers on soft soil foundation. Analyze the multi-source information fusion structure, introduce the least squares support vector machine method to complete the fusion of information on the vibration displacement impact of deep-water pile foundation bridge piers on soft soil foundation. Based on this, select strain measurement points for deep-water pile foundation bridge piers on soft soil foundation, and arrange displacement identification along the entire bridge pier at equal intervals to achieve accurate displacement identification. The experimental results show that the proposed method reduces the error of displacement recognition.

We compute several identities relating curvature actions to second-order differential operators on twistors and sections of related bundles. Based on these formulas, we establish various estimates and vanishing theorems.

In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c² f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years.

Given a tame differential calculus over a noncommutative algebra ${𝒜}$ and an ${𝒜}$-bilinear metric $g_0,$ consider the conformal deformation $g=k.g_0,$$k$ being an invertible element of ${𝒜}.$ We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the metric $g_0.$ As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

There are three well-known Arnold’s invariants $J^{+}$, $J^{-}$, and St (Arnold strangeness) for plane curves. Two of them, $J^{+}$ and $J^{-}$, have been successfully quantized. However, the quantization of the Arnold strangeness St remained unrealized. In this paper, we will express a quantization of St by integrating curvatures multiplied by nontrivial densities. The key here is a partition function by Shumakovitch which reformulates the Arnold strangeness. Our quantized Arnold strangeness includes the rotation number, the original Arnold strangeness, and the Tabachnikov invariants as coefficients in its Taylor expansion.

We study the convergence and curvature blow up of the continuity method on a generalized Hirzebruch surface. We show that the Gromov–Hausdorff convergence is similar to that of the Kähler–Ricci flow and obtain curvature estimates. We also show that a general solution to the continuity method either exists at all times, or the scalar curvature blows up. This behavior is known to be exhibited by the Kähler–Ricci flow.

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

Computer simulations of bi-continuous two-phase fluids with interspersed dumbbells show that, unlike rigid colloids, soft dumbbells do not lead to arrested coarsening. However, they significantly alter the curvature dynamics of the fluid–fluid interface, whose probability density distributions are shown to exhibit (i) a universal spontaneous transition (observed even in the absence of colloids) from an initial broad-shape distribution towards a highly localized one and (ii) super-diffusive dynamics with long-range effects. Both features may prove useful for the design of novel families of soft porous materials.

A study of proper projective symmetry in the Schwarzschild metric is given by using algebric and direct integration techniques. It is shown that projective collineations admitted by the above metric are the Killing vector fields.

A gauge theory of contact is presented, based on the general idea that the local deformation of the nucleon surface at contact should be gauged by the variation of curvature. A contact force is then defined so as to cope with both the variation of curvature and the deformation. This force generalizes the classical definition of surface tension, in that it depends on the mean curvature, but also depends on the variance of the second fundamental form of surface, considered as a statistical variable over the ensemble of contact spots. It turns out that the variance of the second fundamental form does not depend but on the metric of the space of curvature parameters, organized as Riemann space. This result compels us to review the definition of physical surface of a nucleon.

We study the two-dimensional harmonic oscillator on a noncommutative plane. We show that by introducing appropriate Bopp shifts, one can obtain the Hamiltonian of a two-dimensional harmonic oscillator on a sphere according to the Higgs model. By calculating the commutation relations, we show that this noncommutativity is strictly dependent on the curvature of the background space. In other words, we introduce a kind of duality between noncommutativity and curvature by introducing noncommutativity parameters as functions of curvature. Also, it is shown that the physical realization of such model is a charged harmonic oscillator in the presence of electromagnetic field.

In this paper, we study thermodynamic curvature of Chaplygin gas which is one of the dark energy candidates in terms of entropy and temperature. So, we get some information about past universe by considering thermodynamic curvature of Chaplygin gas. In addition to thermodynamic curvature, we study behaviors of test particle in the universe filled with Chaplygin gas.

In this paper, we study the two-dimensional (2D) Euclidean anisotropic Dunkl oscillator model in an integrable generalization to curved ones of the 2D sphere ${𝕊}^2$ and the hyperbolic plane ${ℍ}^2$. This generalized model depends on the deformation parameter $\kappa$ of underlying space and involves reflection operators $R_{x_i}$ in such a way that all the results are simultaneously valid for ${𝕊}^2(\kappa >0)$, ${ℍ}^2(\kappa <0)$ and ${𝔼}^2(\kappa =0)$. It turns out that this system is superintegrable based on the special cases of parameter $\alpha$, which constant measures the asymmetry of the two frequencies in the 2D Dunkl model. Therefore, the Hamiltonian ${{\mathscr{H}}}_{\kappa }$ can be interpreted as an anisotropic generalization of the curved Higgs–Dunkl oscillator in the limit $\alpha \to 0$. When $\alpha =3{\omega }^2$, the system turns out to be the well-known superintegrable 1:2 Dunkl oscillator on ${𝕊}^2$ and ${ℍ}^2$. In this way, the integrals of the motion arising from the anisotropic Dunkl oscillator are quadratic in the Dunkl derivatives for the special cases of $\alpha$. Moreover, these symmetries obtain by the Jordan–Schwinger representation in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical $s{l}_{-1}(2)$ algebra of the 1D Dunkl oscillator. The resulting algebra is a deformation of $s{o}_{\kappa }(3)$ with reflections, which is named the Jordan–Schwinger–Dunkl algebra $js{d}_{\kappa }(3)$. The spectrum of this system is determined by the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

We present an exploration of helical or twisted hypersurfaces family denoted as $x$, which are dependent on three parameters and possess a space-like axis in the Minkowski 4-space ${𝕃}^4$. We provide an in-depth analysis of the fundamental forms, Gauss map, and shape operator associated with $x$. We establish a framework for defining the curvatures of any given family using the Cayley–Hamilton theorem. By applying this theorem, we derive the specific curvatures of the hypersurface under consideration. Furthermore, we investigate the conditions under which the curvatures of $x$ satisfy the umbilical conditions. Lastly, we examine the Laplacian of hypersurfaces family in ${𝕃}^4$, shedding light on its important properties.

In this work, we use the approach recently introduced by Barros to study hadron spectra and some quark confinement properties in a Schwarzchild-like space–time generated by a nongravitational field. As a starting point, for the nongravitational field, we make the choice of a strong Yukawa-like field whose associated potential is a generalized Yukawa-like potential, typical of strong interactions. Then, from the latter field, the energy momentum tensor is constructed, the Einstein field equations are solved and the curvature function of the Schwarzchild metric is obtained. The correspondence principle applied to the Schwarzchild metric has enable us to construct the Dirac equation in the latter space. The resolution of the coupled differential equations of Dirac made it possible to obtain the energy spectrum of the strong interaction. The latter is obtained in a more general form than in the previous investigations. Then, the energy spectrum, masses and confinement radius of few hadrons are estimated and compared with experimental data and other theoretical studies. In most considered cases, our predictions are found to be in good agreement with experimental data. The good agreement observed between our outcomes and the experiment can be attributed to the choice of our potential, which has more free parameters than in past studies with the same approach.

A statistical mechanical model was used to calculate the curvature of the 5 chemically synthesized DNAs which contain repeats sequences (CCTG)_n · (CAGG)_n and (ATTCT)_n · (AGAAT)_n associated with human diseases. 8% polyacrylamide gel analyses were also performed for these 5 DNAs. The results indicate the curvature of the sequences CCTG/bend and ATTCT/bend are larger than that of the sequences CCTG/straight and ATTCT/straight. The curvature of straight/bend is larger than that of CCTG/straight and ATTCT/straight, and smaller than that of CCTG/bend and ATTCT/bend. There exists good consistent between theoretical prediction and experimental data.

Due to the difference of nanotube diameters, the single-walled carbon nanotubes (SWCNTs) show the different energy and mechanical properties. In order to take the effect of the curvature of nanotubes into account in the modeling of those structures, the present paper proposes an atomistic based continuum model with using a type of modified Cauchy-Born to link the continuum strain energy to the interatomic potential. This modified Cauchy-Born is developed by incorporating the concept of differential mean value theorem into the standard Cauchy-Born rule. The present model not only can bridge the microscopic and macroscopic length scales, but also can investigate the curvature effect of a single layer film on the continuum level. Application of the current model to armchair carbon nanotubes and graphite shows an excellent prediction of the size dependent strain energy which are compared in a good agreement with the existing experimental and theoretical results.