Conformal Geometry in Engineering and Medicine

With the development of 3D geometric acquisition technologies, massive 3D geometric data are ubiquitous today. It is of great challenges to process and analyze 3D geometric data efficiently and accurately. Computational conformal geometry is an emerging inter-disciplinary field, which combines modern geometry with computer science and offers rigorous and practical tools for tackling massive geometric data processing problems. The concepts and methods in conformal geometry play fundamental roles in many fields in engineering and medicine.

Conformal geometry studies the invariants under the conformal transformation (angle preserving mapping) group. Conformal geometry is more flexible than Riemannian geometry, and more rigid than topology. Conformal geometry is capable of unifying all shapes in real world to one of three canonical shapes, the sphere, the plane, or the hyperbolic disk; conformal geometric algorithms convert 3D geometric processing problems to 2D image processing problems; furthermore, all surfaces in real life have conformal structures, therefore conformal geometric methods are general. These merits make conformal geometry a powerful tool for real applications.

Theoretical Investigation of New Organic Electroluminescent Materials Based on 4-Azaindole Groups and Oligopyrrole

The electronic properties of four new organic compounds (I-IV) were studied theoretically for application as hole-transporting materials in electroluminescent (EL) devices. We investigated theoretically, the effect of increasing number of pyrrole rings between 4-azainole end moieties. The time dependent density functional theory (TD-DFT/ B3LYP/6-31G(d)) calculated energy gap (E-gap) of the studied compounds decreases in the order of I>II>III>IV; 

the significant reduction of E-gap of compound IV with 2.7 eV compared to 4.27 eV of compound II is due to the bridging effect of C=C(CO2H)2 which remove the steric effect, caused by high dihedral angle between two central pyrrole rings in the non-bridged II. Compound IV posses slow-lying lowest occupied molecular orbital(LUMO) energy levels and low lying highest occupied molecular orbital (HOMO) energy levels, may be promising candidate for hole transporting and bright blue to red emitting layer in organic light emitting device (OLED) fabrication.

Advances in Logic, Operations and Computational Mathematics

Journal of Applied & Computational Mathematics Volume 5, Issue 2 comprised of 7 research articles and 4 opinion articles and is focused on the innovation of polygon, Euler, linear and non-linear equations.

EL-Kholy et al., in their research article discussed about balanced folding over a polygon and Euler numbers. The study proved that for a balanced folding of a simply connected surface M, there is a subgroup of the group which is called all homeomorphisms of M that will acts 1- transitively on the 2-cells of M.

Gil et al., in their research have reported about the exponentially stabile non-linear, non-autonomous multi variable discrete systems. Based on the recent estimates on matrix equations, the findings suggest that a class of non-autonomous discrete-time systems is governed by semi-linear vector difference equations along with slowly varying linear parts.

Models of damped oscillators in quantum mechanics

We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schr¨odinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. 

The factorization technique is applied to the case of a shifted harmonic oscillator. The time evolution of the expectation values of the energy-related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.

Properties of Nilpotent Orbit Complexification

Real and complex nilpotent orbits have received considerable attention in the literature. The former have been studied in a variety of contexts, including differential geometry, symplectic geometry, and Hodge theory.  Also, there has been some interest in concrete descriptions of the poset structure on real nilpotent orbits in specific cases. By contrast, complex nilpotent orbits are studied in algebraic geometry and representation theory — in particular, Springer Theory.

Attention has also been given to the interplay between real and complex nilpotent orbits, with the Kostant-Sekiguchi Correspondence being perhaps the most famous instance. Accordingly, the present article provides additional points of comparison between real and complex nilpotent orbits. Specifically, let g be a finite-dimensional semisimple real Lie algebra with complexification g Each real nilpotent orbit.