Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

We consider the viability problem for nonlinear evolutions inclusions of the form u′(t)?∈?Au(t)?+?F(u(t)), where A is an m-dissipative (possible nonlinear and multi-valued) operator acting in a Banach space X, K?X is a nonempty, locally closed set and $F:K{\user1{ \rightsquigarrow }}X$ is with nonempty, convex, closed and bounded values. We define the concept of A-quasi-tangent set to K at a given point ξ?∈?K and we prove a necessary condition for C ⁰-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the necessary condition is also sufficient. We extend the results to the quasi-autonomous case, we deal with the existence of noncontinuable or even global C ⁰-solutions and, as applications, we deduce a comparison result and a sufficient condition for null controllability.     

Phase-Transfer-Agent-Aided Free Radical Polymerization of Methyl Methacrylate-A Kinetic Study

The kinetics of phase-transfer-agent-aided free radical polymerization of methyl methacrylate was investigated by using K₂S₂O₈ as the initiator and cetyltrimethyl ammonium chloride (Arquad) as the phase transfer agent. The rate of polymerization was found to be proportional to [M]^1.23[K₂S₂O₈]^0.8 [Arquad]^0.25 and almost independent of the volume of water (V _w)/volume of organic solvent (V _o) (benzene) ratio for V _w/V _o < 0.33. A reaction mechanism is proposed.     

On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

On the Chromaticity of Quasi Linear Hypergraphs

An h-hypergraph H is said to be quasi linear if every two edges of H intersect in 0 or r vertices (r ≥ 1 and h ≥ 2r + 1). The main result of this paper is the following: if G and H are chromatically equivalent h-hypergraphs, H is quasi linear and h ≥ 3r ? 1, then any two edges of G intersect in 0,1 or r vertices and the result cannot be strengthened. This extends the corresponding result of the first author for linear hypergraphs (case r = 1) [Tomescu in J Combin Theory B2 72: 229–235, 1988]     

Convolutions of singular measures and applications to the Zakharov system

Uniform L²-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.     

Representation of natural numbers by sums of four squares of integers having a special form

This paper obtains an asymptotic formula for the number of solutions to the equation $ l_1^2 + { }l_2^2 + l_3^2 + l_4^2 = N $ in integers l ₁, l ₂, l ₃, l ₄ such that a < {??l _j } < b, where ?? is a quadratic irrational number, 0 ?? a < b ?? 1, j = 1, 2, 3, 4.