Logarithmic Generalization of the Lambert W Function and Its Applications to Adiabatic Thermostatistics of the Three-Parameter Entropy

A generalization of the Lambert W function called the logarithmic Lambert function is introduced and is found to be a solution to the thermostatistics of the three-parameter entropy of classical ideal gas in adiabatic ensembles. The derivative, integral, Taylor series, approximation formula, and branches of the function are obtained. The heat functions and specific heats are computed using the “unphysical” temperature and expressed in terms of the logarithmic Lambert function.

Squaring Technique using Vedic Mathematics

Vedic Mathematics provides an interesting approach to modern computing applications by offering an edge of time and space complexities over conventional techniques. Vedic Mathematics consists of sixteen sutras and thirteen sub-sutras, to calculate problems revolving around arithmetic, algebra, geometry, calculus and conics. These sutras are specific to the decimal number system, but this can be easily applied to binary computations. This paper presented an optimised squaring technique using Karatsuba-Ofman Algorithm, and without the use of Duplex property for reduced algorithmic complexity. This work also attempts Taylor Series approximation of basic trigonometric and inverse trigonometric series. The advantage of this proposed power series approximation technique is that it provides a lower absolute mean error difference in comparison to previously existing approximation techniques.

Search of weights in the problem of finite-rank signal estimation in presence of noise

The problem of weighted finite-rank time-series approximation is considered for signal estimation in “signal plus noise” model, where the inverse covariance matrix of noise is (2p+1)-diagonal. Finding of weights, which improve the estimation accuracy, is examined. An effective method for the numerical search of the weights is constructed and proved. Numerical simulations are performed to study the improvement of the estimation accuracy for several noise models.

Practical Considerations with Radar Data Height and Great Circle Distance Determination

Owing to their ease of use, “simplified” propagation models, like the Equivalent Earth model, are commonly employed to determine radar data locations. With the assumption that electromagnetic rays follow paths of constant curvature, which is a fundamental assumption in the Equivalent Earth model, propagation equations that do not depend upon the spatial transformation that is utilized in the Equivalent Earth model are derived. This set of equations provides the true constant curvature solution and is less complicated, conceptually, as it does not depend upon a spatial transformation. Moreover, with the assumption of constant curvature, the relations derived herein arise naturally from ray tracing relations.Tests show that this new set of equations is more accurate than the Equivalent Earth equations for a “typical” propagation environment in which the index of refraction n decreases linearly at the rate dn/dh = -1/4a, where h is height above ground and a is the Earth’s radius. Moreover, this new set of equations performs better than the Equivalent Earth equations for an exponential reference atmosphere, which provides a very accurate representation of the average atmospheric n structure in the United States. However, with this n profile the equations derived herein, the Equivalent Earth equations, and the relation associated with a flat Earth constant curvature model produce relatively large height errors at low elevations and large ranges.Taylor series approximations of the new equations are examined. While a second-order Taylor series approximation for height performs well under “typical” propagation conditions, a convenient Taylor series approximation for great circle distance was not obtained.