Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

Analysis of Superradiation Stability of Kerr-Newman Black Hole Using Curve integral

In this article, a new variable y is added here to expand the results of the above article.We use the properties of the Laurent series and the Cauchy integral. When y is greater than a certain limit, the effective potential of the equation does not have a pole, then there is no potential well outside the event horizon, when p 2(a 2 + Q2)/r2 + < ω < mΩH + qΦH,so the Kerr-Newman black hole is superradiantly stable at that time.

Analysis of Superradiation Stability of Kerr-Newman Black Hole Using Curve integral

In this article, a new variable y is added here to expand the results of the above article.We use the properties of the Laurent series and the Cauchy integral. When y is greater than a certain limit, the effective potential of the equation does not have a pole, then there is no potential well outside the event horizon, when p 2(a 2 + Q2)/r2 + < ω < mΩH + qΦH,so the Kerr-Newman black hole is superradiantly stable at that time.

Analysis of Superradiation Stability of Kerr-Newman Black Hole Using Curve integral

In this article, a new variable y is added here to expand the results of the above article.We use the properties of the Laurent series and the Cauchy integral. When y is greater than a certain limit, the effective potential of the equation does not have a pole, then there is no potential well outside the event horizon, when $\sqrt{2(a^2+Q^2)}/{r^2_+}&lt; \omega&lt; m\varOmega_H+q\varPhi_H$,so\ the Kerr-Newman black hole is superradiantly stable at that time.

Coefficients Calculation of Laurent Series in the Neighborhood of Complex Variable Function Poles

The theory of complex function is a key part of mathematics, which can solve the complex problems in production and life. It is of great significance to extend the research field of complex function theory. In this paper, taking a complex variable function as the research object, a calculation method of Laurent series coefficient of complex function pole neighborhood expansion was proposed to determine the complex variable function pole, determine the order of complex variable function pole, calculate the residue of high-order pole in complex variable function, thus judging the attribute of complex variable function. In this regard, the coefficient formula was used to calculate the coefficients of Laurent series in the neighborhood of the complex variable function poles.

Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field ℂ ⁢ ( ( x , y ) ) {\mathbb{C}((x,y))} of Laurent series in two variables over the complex numbers and over function fields of curves over ℂ ⁢ ( ( t ) ) {\mathbb{C}((t))} . We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

Laurent-Hua Luogeng Series with Respect to the Matrix Ball from Space Cn [m × m]

The aim of this work is to obtain multidimensional analogs of the Laurent series with respect to the matrix ball from space Cn [m × m]. To do this, we first introduce the concept of a "layer of the matrix ball" from Cn [m × m], then in this layer of the matrix ball, we use the properties of integrals of the Bochner-Hua Luogeng type to obtain analogs of the Laurent ser