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LetRbe a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse inR. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ∗-ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index inR. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ∗-ring. Some equivalent characterizations for elements inRto be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited. [ABSTRACT FROM AUTHOR]

Geometrical insights are very helpful when learning about the Fourier series. Instead of focusing on trigonometric or exponential functions, we consider a more general collection of orthonormal functions. We show in an elementary way that a Taylor series can also be considered a special case of the generalized Fourier series, with an orthonormal base of power functions and a properly defined inner product. [ABSTRACT FROM AUTHOR]

Approximately all of the petroleum reservoirs are heterogeneous, and this heterogeneity exists on all levels from pores to simulation models. Since reservoir data are available on different scales and each property does not measure at the individual level, scale-up must be used for acquiring properties of coarse grid models. One of the most important reservoir properties in multiphase upscaling is relative permeability. Pseudo functions are the main methods for upscaling of relative permeability. Kyte and Berry and transmissibility weighted pseudos are two kinds of dynamic pseudo functions. The Kyte and Berry method is based on Darcy's law and the transmissibility weighted pseudos concentrate on saturation effects rather than pressure effects and uses a transmissibility weighted average for calculation of pseudo relative permeabilities. In this work, the effects of directional inter-block pseudos and well pseudos were first investigated. Then the results of coarse grid simulation with directional transmissibility weighted pseudos and Kyte and Berry pseudos were compared with the reference fine grid results. [ABSTRACT FROM PUBLISHER]