Differential equations to describe elasticity are derived without the use of stress or strain. The points within the body are the independent parameters instead of strain and surface forces replace stress tensors. These differential equations are a continuous analytical model that can then be solved using any of the standard techniques of differential equations. Although the equations do not require the definition stress or strain, these quantities can be calculated as dependent parameters. This approach to elasticity is simple, which avoids the need for multiple definitions of stress and strain, and provides a simple experimental procedure to find scalar representations of material properties in terms of the energy of deformation. The derived differential equations describe both infinitesimal and finite deformations.

Abstract:
The equations of Euler-Lagrange elasticity describe elastic deformations
without reference to stress or strain. These equations as previously published
are applicable only to quasi-static deformations. This paper extends these
equations to include time dependent deformations. To accomplish this, an
appropriate Lagrangian is defined and an extrema of the integral of this
Lagrangian over the original material volume and time is found. The result is a
set of Euler equations for the dynamics of elastic materials without stress or
strain, which are appropriate for both finite and infinitesimal deformations of
both isotropic and anisotropic materials. Finally, the resulting equations are
shown to be no more than Newton's Laws applied to each infinitesimal volume of
the material.

Abstract:
Linear algebra provides insights into the description of elasticity without stress or strain. Classical descriptions of elasticity usually begin with defining stress and strain and the constitutive equations of the material that relate these to each other. Elasticity without stress or strain begins with the positions of the points and the energy of deformation. The energy of deformation as a function of the positions of the points within the material provides the material properties for the model. A discrete or continuous model of the deformation can be constructed by minimizing the total energy of deformation. As presented, this approach is limited to hyper-elastic materials, but is appropriate for infinitesimal and finite deformations, isotropic and anisotropic materials, as well as quasi-static and dynamic responses.

Abstract:
We compute numerically eigenvalues and eigenfunctions of the quantum Hamiltonian that describes the quantum mechanics of a point particle moving freely in a particular three-dimensional hyperbolic space of finite volume and investigate the distribution of the eigenvalues.

Abstract:
We investigate the numerical computation of Maass cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed r=40000. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the 130millionth eigenvalue.

Abstract:
The author designed a family of nonlinear static electric-springs. The nonlinear oscillations of a massively charged particle under the influence of one such spring are studied. The equation of motion of the spring-mass system is highly nonlinear. Utilizing Mathematica [1] the equation of motion is solved numerically. The kinematics of the particle namely, its position, velocity and acceleration as a function of time, are displayed in three separate phase diagrams. Energy of the oscillator is analyzed. The nonlinear motion of the charged particle is set into an actual three-dimensional setting and animated for a comprehensive understanding.

Abstract:
The electronic structure, energy band structure, total density of states (DOS) and electronic density of perovskite SrTiO_{3} in the cubic phase are calculated by the using full potential-linearized augmented plane wave (FP-LAPW) method in the framework density functional theory (DFT) with the generalized gradient approximation (GGA) by WIEN2k package. The calculated band structure shows a direct band gap of 2.5 eV at the Γ point in the Brillouin zone.The total DOS is compared with experimental x-ray photoemission spectra. From the DOS analysis, as well as charge-density studies, I have conclude that the bonding between Sr and TiO_{2} is mainly ionic and that the TiO_{2} entities bond covalently.The calculated band structure and density of state of SrTiO_{3} are in good agreement with theoretical and experimental results.

Abstract:
Hoyle and
Narlikar (HN) in the 1960’s [1]-[3] developed a theory of gravitation which was completely Machian and used
both retarded and advanced waves to communicate gravitational influence between
particles. The advanced waves, which travel backward in time, are difficult to
visualize and although they are mathematically allowed by relativistic wave
equations, they never really caught on. The HN theory reduced to Einstein’s
theory of gravity in the smooth fluid approximation and a transformation into
the rest frame of the fluid. Hawking [4] in 1965 pointed out a possible flaw in the theory. This involved
integrating out into the distant future to account for all the advanced waves
which might influence the mass of a particle here and now. Hawking used
infinity as his upper time limit and showed the integral was divergent. We
point out that since the universe is known to be expanding, and accelerating,
the upper limit in the advanced wave time integral should not be infinite but is
bounded by the Cosmic Event Horizon. This event horizon H_{e} represents a barrier between future events that can
be observed and those which cannot. We show that the advanced wave integral is
finite when H_{e}/C, is used as the upper limit of the
advanced wave integral. Hawking’s objection is no longer valid and the HN
theory becomes a working theory once again.

Abstract:
Rare earth elements (REE) in sedimentary rocks are most suitable for source rock characterization. Rare earth element data of the sandstones of the unmetamorphosed Meso-Neoproterozoic Chhattisgarh and Indravati basins and the metamorphosed Paleoproterozoic Sakoli and Saucer basins of the Bastar craton have been studied for source rock characterization. The quartzites have higher ∑R EE mean value (145 ppm) compared to the sandstones (34 ppm). The REE patterns of all the three formations of the Chandarpur Group of the Chhattisgarh basin and the Tiratgarh Formation of the Indravati basin are uniform and there are no systematic differences in REE patterns among different formations of the Chandarpur Group and the Tiratgarh Formation. The REE patterns of the quartzites are similar to the REE patterns of the sandstones. Chondrite-normalized REE patterns with LREE enrichment and a strong negative Eu anomaly of the sandstones and quartzites gives a broad hint about felsic source rocks. The source rocks are identified as Archean granite and gneiss of the Bastar craton. The REE mixing modeling of the sandstones and quartzites suggest that the exposed the Proterozoic upper crust of the Bastar craton during the sedimentation of the Paleoproterozoic Sakoli and saucerand the Meso-Neoproterozoic Chhattisgarh and Indravati basins of the Bastar craton was largely consisted of gneissic rocks (70%), with a little contribution (20%) from Late Archean (2.5 Ga) granites. The present study does not suggest any significant change in the upper crustal composition during Proterozoic in the Bastar craton.