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Social network algorithmics

We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model

Above-threshold detachment of electrons from negative ions by a bicircular laser field is analyzed within the strong-field approximation. A bicircular field consists of two counter-rotating circularly polarized fields with angular frequencies rω and sω with integer r and s. The energy and angle resolved spectra of the detached electron are invariant with respect to a rotation by the angle 360°/(r+s). The results obtained are explained in terms of the interference of contributions to the T matrix from different complex solutions of the saddle-point equation.

https://iopscience.iop.org/article/10.1002/lapl.200610119/pdf

Above-threshold detachment by a two-color bicircular laser field

In this paper we consider the-stabilization of th-order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n + 1 and at least using an integral factorization of the corresponding characteristic function. A necessary condition for-stabilizability is analyzed utilizing the property that the derivative of \alpha-stable quasipolynomial is also-stable under certain conditions. Sufficient and necessary conditions are given for systems with real-rooted open loop characteristic function: the delay intervals are determined where the conditions for dominancy and \alpha-stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multi-link inverted pendulum.

/pdf/conditions-for-stabilizability-of-time-delay-systems-with-xwgvimu1kc.pdf

Conditions for stabilizability of time-delay systems with real-rooted plant

Four. Functions of One Complex Variable. Special Part.- Five. The Location of Zeros.- Six. Polynomials and Trigonometric Polynomials.- Seven. Determinants and Quadratic Forms.- Eight. Number Theory.- Nine. Geometric Problems.- x 1 Additional Problems to Part One.- New Problems in English Edition.- Author Index.- Topics.

Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry

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Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry