A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force

This work suggested a new generalized fractional derivative which is producing different kinds of singular and nonsingular fractional derivatives based on different types of kernels. Two new fractional derivatives, namely Yang-Gao-Tenreiro Machado-Baleanu and Yang-Abdel-Aty-Cattani based on the nonsingular kernels of normalized sinc function and Rabotnov fractional-exponential function are discussed. Further, we presented some interesting and new properties of both proposed fractional derivatives with some integral transform. The coupling of homotopy perturbation and Laplace transform method is implemented to find the analytical solution of the new Yang-Abdel-Aty-Cattani fractional diffusion equation which converges to the exact solution in term of Prabhaker function. The obtained results in this work are more accurate and proposed that the new Yang-Abdel-Aty-Cattani fractional derivative is an efficient tool for finding the solutions of other nonlinear problems arising in science and engineering.     

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0.5−?) for any ?>0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0.5−?) for any ?>0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.     

Design of Controllable Batch Processes in the Presence of Uncertainty

Design of controllable batch processes can be more challenging than continuous processes because of their unsteady nature of operation. The operating strategy of a batch process is characterized by trajectories of manipulated variables. This precludes the use of conventional controllability measures in evaluating the controllability of a given batch process design. Short process development cycles typically accredited to batch processes lead to uncertainty in the model formulation. Integrated approach to batch process design and control addresses the problem of controllability of a batch process during the design phase. This is best achieved by treating the problem as a dynamic optimization problem with time invariant (design) and time variant (operating) variables. The method proposed in this paper uses the decomposition feature of Generalized Benders Decomposition (GBD) to evolve a 2-level nested optimization problem (primal and master), one involving time variant decision (operating) variables and the other involving time invariant decision (design) variables. To enhance the computational efficiency, a relaxed LP formulation of the master problem is proposed. This variant of GBD, termed as ExGBD, is guaranteed to converge to the optimum for convex problems. A simple batch reactor design problem has been chosen to demonstrate ExGBD.     

Laser-polarized Xe NMR and MRI at Ultralow Magnetic Fields

Laser-polarized ¹²⁹Xe and a high-T_csuperconducting quantum interference device (SQUID) are used to obtain magnetic resonance images in porous materials at a magnetic field of 2.3 mT, corresponding to a Larmor frequency of 27 kHz. Image resolution of 1 mm is obtained with gradients of only 1 mT/m. The resolution of xenon chemical shifts in different physicochemical environments at ultralow fields is also demonstrated. Details of the circulating flow optical pumping apparatus and the SQUID spectrometer are presented.     

Assessment of branching in polymers from small-angle x-ray scattering (SAXS)

Small-angle x-ray scattering from solutions was applied in assessing branching using well-characterized dextran hydrolyzates. It was found that the longitudinal and cross-sectional properties of the polysaccharides vary in a manner predictable by the molecular topology of the chains. A hydrodynamically equivalent linear model for dextran is proposed and it has been shown that this model correctly predicts values of Φ and K, Flory's universal hydrodynamic constant and the Flory–Fox–Schaefgen parameter, respectively.     

Model for lattice dynamics in metals

A lattice dynamical model for cubic metals, which satisfies the internal force equilibrium condition of the lattice, is proposed. The present model combines a linearized Thomas Fermi theory for the electron-ion interaction and the axially symmetric model for the ion-ion interaction. The computed frequencies of copper and sodium are in good agreement with the experimental data.     

Measurements of line intensities in the two-micron band of ammonia

Intensities of about four hundred lines of ammonia in the 2μ region have been measured using Doppler-broadened lines. Comparison with rigid-rotor calculations gives fair agreement between theory and experiment. The total integrated intensity of the ν₃ + ν₄ (perpendicular) band has been estimated to be 17.19 cm^-2 atm^-1 at 296 K.     

Measurements of intensities and nitrogen-broadened linewidths in the CO fundamental at low temperatures

Intensities and nitrogen-broadened half-widths of lines R(0), R(8) and R(16) in the fundamental band of ¹²C¹⁶O have been measured at 83°K, 100°K, 150°K, 200°K and 298°K. The intensities of several other lines in the P- and R-branches of the band have also been measured at 298°K. The absolute intensity derived from the line intensity data using the Herman-Wallis formula is S^°_v = 273 ± 10 cm^-2atm^-1 at S.T.P. A separate measurement employing the Wilson-Wells-Penner-Weber method has yielded S^°_v = 277 ± 4 cm^-2 atm^-1 at S.T.P. Both of these values are within 6 per cent of most of the previously published direct measurements of this parameter. The values for the line intensities reported earlier by other authors are lower by nearly 16 per cent.     

The Steiner tree problem II: Properties and classes of facets

This is the second part of two papers addressing the study of the facial structure of the Steiner tree polyhedron. In this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the Steiner tree problem is known to be NP-hard.Corresponding author.The author appreciates partial support from National Science Foundation Grants Nos. DSM-8606188 and ECS 8800281.