In the present paper‎, ‎we propose an iterative algorithm for‎ ‎solving the generalized $(P,Q)$-reflexive solution of the quaternion matrix‎ ‎equation $overset{u}{underset{l=1}{sum}}A_{l}XB_{l}+overset{v} ‎{underset{s=1}{sum}}C_{s}widetilde{X}D_{s}=F$‎. ‎By this iterative algorithm‎, ‎the solvability of the problem can be determined automatically‎. ‎When the‎ ‎matrix equation is consistent over...

S. Prešić ([6]) determined the all reproductive general solutions of the equation r(x) = 1, where r was a unary relation of a given set T . In this paper we determine the all general solutions of this equation. S. Prešić ([6]) introduced the most general concept of equations. Namely, he considered the equation of the form r(x) = 1, where r was a unary relation of a non–empty set S. He gave the ...

The gauge equivalence between the (2+1)-dimensional Zakharov equation and (2+1)-dimensional integrable continuous Heisenberg ferromagnetic model is established. Also their integrable reductions are shown explicitly. Preprint CNLP-1994-04. Alma-Ata.1994 The concepts of gauge equivalence between completely integrable equations plays important role in the theory of solitons[1,2]. In the (2+1)-dime...

We consider the loop equation in four-dimensional N = 4 SYM, which is a functional differential equation for the Wilson loop W (C) and expresses the propagation and the interaction of the string C. Our W (C) consists of the scalar and the gaugino fields as well as the gauge field. The loop C is specified by six bosonic coordinates y(s) and two fermionic coordinates ζ(s) and η(s) besides the fou...

It is proved that every measurable, non-vanishing cocycle defined on the product of (0,∞) and an arbitrary compact metric space is continuous. Some other sufficient conditions for continuity of a cocycle are also given. Consider functions F satisfying the translation equation F (s+ t, x) = F (t, F (s, x)) (T) and real or complex valued solutions of the equation G(s+ t, x) = G(s, x)G(t, F (s, x)...

We present a local sensitivity analysis for the kinetic Cucker-Smale (C-S) equation with random inputs. This is a companion work to our previous local sensitivity analysis for the particle C-S model. Random imputs in the coefficients of the kinetic C-S equation can be caused by diverse sources such as the incomplete measurement and interactions with unknown environments, and will enter the prob...

Transmission ellipsometry measures the real and imaginary parts of the ratio τ = t(p)/t(s), where t(s), and t(s) are the transmission amplitudes for thep and s polarizations. For a homogeneous layer, the unknowns to be determined are the layer dielectric constant ε = n(2) and the layer thickness Δz. For nonabsorbing films the thickness can be eliminated, and an algebraic equation for e results....

The possibility is discussed to improve the ray approximation up to an exact representation of a wave field by the Feynman–Kac probabilistic formula (this formula gives an exact solution of the Helmholtz equation in the form of the expectation of a certain functional on the space of Brownian random walks). Some examples illustrate an application of the solutions obtained to diffraction problems...

In this paper, we study the computability of the solution operator of the initial problem for the nonlinear Kawahara equation, which is based on the Type-2 Turing machines. We will prove that in Sobolev space 2 ( ) s H R , for 0 s ≥ , the solution operator: 2 : ( ) s R K H R → ( ; C R 2 ( )) s H R is ( ,[ s H δ ρ → ]) s H δ -computable. The conclusion enriches the theory of computability. Key w...

In this paper, an iterative method is proposed for solving matrix equation $sum_{j=1}^s A_jX_jB_j = E$. This method is based on the global least squares (GL-LSQR) method for solving the linear system of equations with the multiple right hand sides. For applying the GL-LSQR algorithm to solve the above matrix equation, a new linear operator, its adjoint and a new inner product are dened. It is p...