Fundamental Tensor Research Articles

Plane gravitational waves of gauge-invariant generalized field equations with asymmetric fundamental tensor in plane symmetry

Buchdahl [Q. J. Math. 8, 89 (1957) and 9, 257 (1958)] has proposed the study of gauge-invariant generalization of field theories with asymmetric fundamental tensor. In this paper, the solution of plane gravitational waves in these Buchdahl's field theories have been investigated by the authors in plane symmetric space-time. Plane symmetry was studied and defined by Taub [Ann. Math. 53, 472 (1951)]. It has been shown that these solutions become identical with those of strong field equations of Einstein’s nonsymmetric unified field theory in plane symmetry under certain cases.

Symmetrical Fundamental Tensors, Differential Operators, and Integral Theorems in Differential Geometry

To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of networks of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reasonable and natural results of the symmetrical analytical system.

Thermodynamically Consistent Anisotropic Plasticity Model

AbstractThe objective of this paper is to consider the thermodynamically consistent anisotropic plasticity model based on full decomposition of stress tensor into generalized deviatoric part and generalized spherical part of stress tensor. Two fundamental tensors αij and βij, which represent anisotropic material properties, are defined and can be considered as generalizations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors αij and βij, the concept of total generalized “pressure” and pressure corresponding to the volumetric deformation is redefined. It is shown that the formulation of anisotropic plasticity in the case of incompressible plastic flow must be considered independently from the generalized hydrostatic pressure. Accordingly, a modification to the anisotropic Hill criterion is introduced. Based on experimental research, which has been published, the modified Hill (1948, “A Theory of the Yielding and Plastic Flow of Anisotropic Metals,” Proc. R. Soc. London, Ser. A, 193(1033), pp. 281–297;1950, Mathematical Theory of Plasticity, Clarendon, Oxford) criterion for anisotropic elastoplasticity is validated. The results are presented and discussed, and future studies are outlined.

Experimental and Computational Characterization of the 17O Quadrupole Coupling and Magnetic Shielding Tensors for p-Nitrobenzaldehyde and Formaldehyde

We have used solid-state 17O NMR experiments to determine the 17O quadrupole coupling (QC) tensor and chemical shift (CS) tensor for the carbonyl oxygen in p-nitro-[1-(17)O]benzaldehyde. Analyses of solid-state 17O NMR spectra obtained at 11.75 and 21.15 T under both magic-angle spinning (MAS) and stationary conditions yield the magnitude and relative orientation of these two tensors: CQ = 10.7 +/- 0.2 MHz, etaQ = 0.45 +/- 0.10, delta11 = 1050 +/- 10, delta22 = 620 +/- 10, delta33 = -35 +/- 10, alpha = 90 +/- 10, beta = 90 +/- 2, gamma = 90 +/- 10 degrees. The principal component of the 17O CS tensor with the most shielding, delta33, is perpendicular to the H-C=O plane, and the tensor component with the least shielding, delta11, lies along the C=O bond. For the 17O QC tensor, the largest (chi(zz)) and smallest (chi(xx)) components are both in the H-C=O plane being perpendicular and parallel to the C=O bond, respectively. This study represents the first time that these two fundamental 17O NMR tensors have been simultaneously determined for the carbonyl oxygen of an aldehyde functional group by solid-state 17O NMR. The reported experimental solid-state 17O NMR results provide the first set of reliable data to allow evaluation of the effect of electron correlation on individual CS tensor components. We found that the electron correlation effect exhibits significant influence on 17O chemical shielding in directions within the H-C=O plane. We have also carefully re-examined the existing experimental data on the 17O spin-rotation tensor for formaldehyde and proposed a new set of best "experimental" 17O chemical shielding tensor components: sigma11 = -1139 +/- 80, sigma22 = -533 +/- 80, sigma33 = 431 +/- 5, and sigma(iso) = -414 +/- 60 ppm. Using this new set of data, we have evaluated the accuracy of quantum chemical calculations of the 17O CS tensors for formaldehyde at the Hartree-Fock (HF), density-functional theory (DFT), Møller-Plesset second-order perturbation (MP2), and coupled-cluster singles and doubles (CCSD) levels of theory. The conclusion is that, while results from HF and DFT tend to underestimate the electron correlation effect, the MP2 method overestimates its contribution. The CCSD results are in good agreement with the experimental data.

A modification of Einstein–Schrödinger theory that contains both general relativity and electrodynamics

We modify the Einstein-Schrodinger theory to include a cosmological constant $\Lambda_z$ which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant $\Lambda_z$ is assumed to be nearly cancelled by Schrodinger's cosmological constant $\Lambda_b$ which multiplies the nonsymmetric fundamental tensor, such that the total $\Lambda=\Lambda_z+\Lambda_b$ matches measurement. The resulting theory becomes exactly Einstein-Maxwell theory in the limit as $|\Lambda_z|\to\infty$. For $|\Lambda_z|\sim 1/(Planck length)^2$ the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are $<10^{-16}$ of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein-Infeld-Hoffmann (EIH) equations of motion match the equations of motion for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein-Schrodinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner-Nordstrom solution except for additional terms which are $\sim 10^{-66}$ of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein-Maxwell theory.

The boundary element method applied to forced convection heat problems

An integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier–Stokes and energy equations are used for calculation of the flow and temperature fields. The governing differential equations, in terms of primitive variables, are derived using velocity–pressure–temperature parameters. The calculation of fundamental solutions and solutions tensor is shown. Applications to simple flow cases, such as driven cavity, forward facing step, deep cavity and channel are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers.

On the Stability in Bonnet's Theorem of the Surface Theory

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 &gt; 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

On H-Projectively flat KH-Structure submanifold

In this paper we have obtained the condition for a submanifold of a KH-structure manifold to be KH-structure. A few relations are found in the second fundamental tensor H and k. Using these results we have also find out the flatness of H-projective curvature tensor in KH-structure submanifold.

On Holomorphic Curvature of Complex Finsler with special (α, β)−Metric

The notion of the holomorphic curvature for a Complex Finsler space (M, F) is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. This paper is about the fundamental metric tensor, inverse tensor and as a special approach of the pull-back bundle is devoted to obtain the Riemannian curvature and holomorphic curvature of Complex Finsler with special (α, β)-metric

A Method to Evaluate the Elastic Properties of Ceramics-Enhanced Composites Undertaking Interfacial Delamination

A micromechanical approach is developed to investigate the behavior of composite materials, which undergo interfacial delamination. The main objective of this approach is to build a bridge between the intricate theories and the engineering applications. On the basis of the spring-layer model, which is useful to treat the interfacial debonding and sliding, the present paper proposes a convenient method to assess the effects of delamination on the overall properties of composites. By applying the Equivalent Inclusion Method (EIM), two fundamental tensors are derived in the present model, the modified Eshelby tensor, and the compliance tensor (or stiffness tensor) of the weakened inclusions. Both of them are the fundamental tensors for constructing the overall constitutive law of composite materials. By simply substituting these tensors into an existing constitutive model, for instance, the Mori-Tanaka model, one can easily evaluate the effects of interfacial delamination on the overall properties of composite materials. Therefore, the present method offers a pretty convenient tool. Some numerical results are carried out in order to demonstrate the performance of this model.

THE GEOMETRY OF THE TORUS UNIVERSE

In this contribution, we show that the cyclic universe models naturally emerge from torus geometry in a braneworld scenario. The Riemannian metric on torus and the fundamental tensors of the General Relativity are derived. A discussion on particular aspects of this model is also given.

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α ij and β ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α ij and β ij , the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.

Probabilistic Modeling for Composites with Particle Interface Degrading

A popular type is of metal matrix reinforced by ceramics particles. It has been revealed that the composites are susceptible to interfacial degrading, which may be dramatically detrimental to the overall properties of composites. In present paper, an equivalent inclusion type constitutive model was developed for the composites with dispersed particles on which imperfections of the material interface are incurred. Two fundamental tensors are derived, the modified Eshelby tensor and the damage tensor of the weakened particles. By applying these tensors into a carefully schemed constitutive law, the effects of interfacial degrading on the overall properties of composite materials can be investigated. The interface degrading includes sliding and debonding. The numerical results show that even with the nil resistible sliding coefficient, its effect on the overall Young’s modulus is not notable unless the volume fraction of the particle is so high as more than 70%. For the global Poisson’s ratio, when there is the sliding on the interface, the Poisson’s ration rises irrespective of the constituent material values. It is noted that even in the elastic state, the global Poisson’s ratio rises greater than that of both the constituents. This phenomenon might indicate that even at the elastic state, the particle interfacial sliding would give somewhat a plasticity-like deformation behavior. The effect of the interfacial debonding on the overall properties of the composite is more conspicuous in comparisons of the sliding. The debonding parameter greatly affects both the properties for almost entire range of the particle volume fraction. Unlike the sliding effect case, the debonding decreases Poisson’s ratio at all cases, which represents the micro-damage effect occurring in the composite.

Running neutrino masses and mixing in a SU(4)×SU(2)2 × U(1)X model

In this talk, we discuss the implications of the renormalization group equations for the neutrino masses and mixing angles in a supersymmetric string-inspired SU(4)×SU(2)L×SU(2)R×U(1)X model with matter in fundamental and antisymmetric tensor representations only. The quark, charged lepton and neutrino Yukawa matrices are distinguished by different Clebsch-Gordan coefficients due to contracting over SU(4) and SU(2)R indices. In order to permit for a more realistic, hierarchical light neutrino mass spectrum with bi-large mixing a second U(1)X breaking singlet with fractional charge is introduced. By numerical investigation we find a region in the model parameter space where the neutrino mass-squared differences and mixing angles at low energy are consistent with experimental data.

Recovering the inflationary potential and primordial power spectrum with a slow roll prior:methodology and application to WMAP three year data

We introduce a new method for applying an inflationary prior to a cosmological data setthat includes relations between observables at arbitrary order in the slow roll expansion.The process is based on the inflationary flow equations, and the slow roll parametersappear explicitly in the cosmological parameter set. We contrast our method with otherways of imposing an inflationary prior on a cosmological data set, and argue that thismethod is ideal for use with heterogeneous data sets. In particular, it would bewell suited to exploiting any direct detection of fundamental tensor modes by aBBO-style mission. To demonstrate the practical use of this method we apply it to theWMAPI+All data set, and the newly released WMAPII data set on its own and together with the SDSSdata. We find that all basic classes of single field inflationary models are still allowed at the1−2σ level,but the overall parameter space is sharply constrained. In particular, we find evidence that the combinationWMAPII+SDSS is sensitive to effects arising from terms that are quadratic in the two leading order slowroll parameters.

Information measures based on Tsallis’ entropy and geometric considerations for thermodynamic systems

An analysis of the thermodynamic behavior of quantum systems can be performed from a geometrical perspective investigating the structure of the state space. We have developed such an analysis for nonextensive thermostatistical frameworks, making use of the q -divergence derived from Tsallis’ entropy. Generalized expressions for operator variance and covariance are considered, in terms of which the fundamental tensor is given.

On curvature characterizations of some hypersurfaces in spaces of constant curvature

We investigate curvature properties of pseudosymmetry type of hypersurfaces in semi-Riemannian spaces of constant curvature having the minimal polynomial for the second fundamental tensor of third degree. Among other things we show that the curvature tensor of such hypersurfaces satisfies some condition, which is a generalization of the Roter type equation.

The boundary element method applied to incompressible viscous fluid flow

An Integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier-Stokes and energy equations are used for calculation of the flow field. The governing differential equations, in terms of primitive variables, are derived using velocity-pressure-temperature. The calculation of fundamental solutions and solutions tensor is showed. Applications to simple flow cases, such as the driven cavity, step, deep cavity and channel of multiple obstacles are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers.

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes (Part II): Applications

Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss curvature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.

Fermion masses and proton decay in string-inspired [formula omitted

We present a supersymmetric model of fermion masses with SU(4)×SU(2)2×U(1)X gauge group with matter in fundamental and antisymmetric tensor representations only. The up, down, charged lepton and neutrino Yukawa matrices are distinguished by different Clebsch–Gordan coefficients due to contracting over SU(4) and SU(2)R indices. We obtain a hierarchical light neutrino mass spectrum with bi-large mixing. The condition that anomalies be cancelled by a Green–Schwarz mechanism leads to fractional U(1)X charges which exclude B violation through dimension-4 and -5 operators.