ON THE SOLVABILITY OF THE NULLATOR-NORATOR PAIRS NETWORK

The paper deals with the unique solvability of a nullator-norator pairs network consisting of RLC elements and source generators. After defining the kernel of a network the normaL inverse normal, distinguished and reactance trees of the network graph are introduced. Setting out from [6] necessary and sufficient conditions are given for the unique solvability. A topological formula is introduced from which many sufficient conditions of the unique solvability can be obtained as algebraic equations between the parameters of the RLC network elements. The results of the paper are illustrated by examples. Finally. a block scheme is presented for the examination of the unique solvability by computer technique

Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for $d=2,3$) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity $u_0\in H^s(\R^2)$ for $s>0$ in 2-D, or $u_0\in H^1(\R^3)$ satisfying |u_0|_{L^2}|\na u_0|_{L^2} being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10], which requires the initial velocity $u_0\in H^2(\R^d)$ for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result

Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and $\mathcal{W}$-property. Based on the unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP)