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In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato–Fefferman–Phong’s class in Lipschitz domains. An elementary proof of the doubling property for u 2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the ℬ p weight properties for the solution u near the boundary. [ABSTRACT FROM AUTHOR]

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra $$W^C_{1+\infty}$$ , which is a subalgebra of W 1+∞. [ABSTRACT FROM AUTHOR]

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈ U( R) such that 1 R ± u ∈ U( R), if and only if for any a∈ R, there exists a u ∈ U( R) such that a ± u ∈ U( R). Furthermore, we prove that, for any A ∈ M n ( R)( n ≥ 2), there exists a U ∈ GL n ( R) such that A ± U ∈ GL n ( R). [ABSTRACT FROM AUTHOR]

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

The article presents a research on the discontinuous quasilinear elliptic variational-hemivariational inequalities which involves the ρ-Laplacian. It also notes on the extension of the theory for discontinuous problems with the use of the method of sub- and supersolutions and compared on the results of S. Carl. It also aims to prove the existence of extremal solutions within a given order interval of sub- and supersolutions. It also gives example of the construction of sub- and supersolutions.

The article presents a research on the iterative methods for generalized von Foerster equations with functional dependence. It examines the meeting of a natural iterative method to the exact solution of a differential-functional von Foerster-type equation that notes a single population depending on its total size, past time and states densities. It also assumes either Perron comparison conditions and some monotonicity. It also investigates on the natural conditions which ensures convergence of iterative methods.