Projective Invariance in Cosmological Models and Dark Energy

We study theories of gravitation that are based on the Einstein – Hilbert action that are not projectively invariant and can therefore completely determine their connections. We are thus lead to the conclusion that the geometry is necessarily Riemann – Cartan and at least the trace part of a torsion field must be present. We examine the consequence of including these torsion fields in cosmological models. Our results differ from those obtained earlier in the Einstein – Cartan – Sciama – Kibble theory. We also consider a model that includes a series of quadratic torsion terms. This series leads to a potential function that has the effect of “turning on” the cosmological constant. This potential function then acts like dark energy. This model also shows that the torsion field can produce an inflationary period. PACS: 04.02 Cv, 95.30 Sf, 98.80-k     

Regular Twists: Replicating Coulomb’s Wire-Torsion Experiments

I discuss our replication of the wire-torsion experiments that Charles Augustin Coulomb (1736–1806) reported in a session of the Paris Académie des Sciences in 1784. I first explain the nature and purpose of the replication method and then apply it to an analysis of Coulomb’s experiments. I conclude by placing Coulomb’s presentation of his memoir into its specific historical contest.     

A Note on Mathematical Relationships Among Bond-Torsion Force Fields

A set of mathematical relationship between torsion potential functions such as trigonometric and Fourier series is presented herein. A harmonic approximation form is also introduced, and its stiffness constant is related to the parameters of trigonometric and Fourier series. Mathematical relationships between various force field parameters are presented in the form of conversion matrices.     

A Cheeger–Müller theorem for delocalized analytic torsion

We study the delocalized L ²-analytic torsion introduced by John Lott, and define the delocalized L ²-combinatorial torsion. By using the method of Bismut–Zhang, under the conditions of positive Novikov–Shubin invariants and finite conjugacy class, we get the Cheeger–Müller type relation between the delocalized L ²-analytic torsion and the delocalized L ²-combinatorial torsion.      

Degenerate Hermitian Manifolds

The geometry of almost complex manifolds with degenerate indefinite Hermitian metrics is studied.     

An explicit formula for the Webster torsion of a pseudo-hermitian manifold and its application to torsion-free hypersurfaces Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.     

Effect of Torsion and Warping on the Free Vibration of Sandwich Beams

The problem on free vibrations of wide sandwich beams is tackled in this paper. Torsional and warping effects in addition to flexure are included in the formulation of the dynamic problem. In order to show the effects of bending-torsion coupling and warping on the natural frequencies and the corresponding vibration modes, three cases are considered. First, the warping and torsion effects are neglected, second, the effect of warping on the coupling terms is neglected, and third, the effect of warping on the coupling terms is included. The viscoelastic core is modeled by elastic translational and rotational springs. The finite-difference method is used to solve the partial differential equations of motion with different boundary conditions for the top and bottom layers. Results for different materials, fiber orientations, depth-to-width ratios, and boundary conditions are found. The natural frequencies and the corresponding vibration modes obtained are in a good agreement with those cited in the literature. If the bending-torsion coupling is pronounced, the inclusion of warping affects the natural frequency considerably.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 163–176, April–May, 2005.     

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

It is shown that, depending upon the orientation of the end tangents t₀,t₁ relative to the end point displacement vector p=p₁–p₀, the problem of G¹ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points p₀,...,p_n in R³, compatible with a G¹ piecewise-PH-cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape-preservation properties of the resulting curves, is illustrated by a selection of computed examples.