We deene an innnite array A of nonnegative integers based on a linear recurrence, which produces basis elements of an exotic ternary numeration system. Using the numeration system we explore many properties of A. Further, we propose and analyze a family of 2-player games on a variable number of heaps of tokens, and present a winning strategy based on certain subarrays of A. Though the strategy ...

This article is a short supplement to our previously published paper, in which we proved that each semialgebraic set can be represented as a projection of a solution set of some system of interval linear equations with dependent coeecients. The new result says that interval occurring can be chosen as narrow as wanted. The new result is proved by a simple linear transformation.

The two coupled linear equations C*X-Z%B = A^ C^Y+Z^A^B^ in polynomial matrices are studied in detail. These equations are crucial in the theory and design of linear optimal dynamic regulators via frequency-domain methods. The solvability of these equations is established under natural conditions. All solutions are characterized and then a specific solution is studied. Relation to other matrix ...

We obtain linearized oscillation theorems for the equation with distributed delays ẋ(t)+ m ∑ k=1 rk(t) ∫ t −∞ fk(x(s)) ds Rk(t, s) = 0. (1) The results are applied to logistic, Lasota–Wazewska and Nicholson’s blowflies equations with a distributed delay. In addition, the “Mean Value Theorem” is proved which claims that a solution of (1) also satisfies the linear equation with a variable concent...

Using inequalities for a certain function appearing in the half-linear version of Picone’s identity, we show that oscillatory properties of the half-linear second-order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|p−2x, p > 1, can be investigated via oscillatory properties of a certain associated second-order linear differential equation. This linear equation plays the role of a...

This paper studies the problem of optimal control of the viscous KdV-Burgers’ equation. We develop a technique to utilize the Cole-Hopf transformation to solve an optimal control problem for the viscous KdV-Burgers’ equation. While the viscous KdV-Burgers’ equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show t...

We describe a general method for enclosing the solution set of a system of interval linear equations. We present a general theorem and an algorithm in a MATLAB-style code. The result is called a “method”, not an “algorithm”, because it involves solving absolute value matrix inequalities; the way how to solve these inequalities will be explained elsewhere.

Analysis of the generalized Weierstrass-Enneper system includes the estimation of the degree of indeterminancy of the general analytic solution and the discussion of the boundary value problem. Several different procedures for constructing certain classes of solutions to this system, including potential, harmonic and separable types of solutions, are proposed. A technique for reduction of the W...

The classical PnP problem is premised on given intrinsic camera parameters. However, for unknown intrinsic camera parameters, given n space points in a world coordinate system and their coordinates in an image coordinate system, the extrinsic camera parameters can be determined. Regarding the existence and uniqueness of a solution for the classical PnP problem, for 4 control points in a plane a...

A new method of designing observer-based deadbeat controllers for linear multivariable systems is presented. The design procedure consists in solving a linear equation in polynomial matrices and yields the entire controller, including the observer plus state feedback, in a single step. The resulting algorithm is remarkably simple, efficient, and leads to an economical realization of the control...