Consumer Optimization and a First-Order PDE with a Non-Smooth System

We study a first-order nonlinear partial differential equation and present a necessary and sufficient condition for the global existence of its solution in a non-smooth environment. Using this result, we prove a local existence theorem for a solution to this differential equation. Moreover, we present two applications of this result. The first concerns an inverse problem called the integrability problem in microeconomic theory and the second concerns an extension of Frobenius’ theorem.

Perron and Frobenius Meet Carathéodory

We present a new approach of proving certain Carathéodory-type theorems using the Perron–Frobenius Theorem, a classical result in matrix theory describing the largest eigenvalue of a matrix with positive entries. At the end, we list some results and conjectures that we hope can be approached with this method.

Doubly stochastic matrices and the quantum channels

The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.

Derivative number apparatus and application possibilities

The article develops the apparatus of derived numbers, the use of which allows one to study the behavior of functions of several variables without requiring their differentiability. In addition, the application of this apparatus to the problem of integrability of the field of planes tangent to a differential manifold allows one to generalize the Frobenius theorem and expand its scope by weakening the restrictions on the degree of smoothness of the differential manifolds under consideration. Conditions and criteria for using the apparatus of partial and external derivatives of numbers are proposed.