Volume 4, Issue 1

We show that there are a cardinal μ , a σ -ideal I ⊆ P ( μ ) and a σ -subalgebra B of subsets of μ extending I such that B/I satisfies the c.c.c. but the quotient algebra B/I has no lifting.

It is known that the following two fundamental properties of porosity fail for symmetric porosity: 1) Every nowhere dense set A contains a residual subset of points x at which A has porosity 1. 2) If A is a porous set and 0 < p < 1, then A can be written as a countable union of sets, each of which has porosity at least p at each of its points . Here we explore the somewhat surprising extent to which these properties fail to carry over to the symmetric setting and investigate what symmetric analogs do hold.

In this note we will construct several additive Darboux-like functions ƒ : ℝ → ℝ answering some problems from (an earlier version of) [Gibson, Natkaniec, Real Anal. Exchange, 22: 492–533, 1996–97]. In particular, in Section 2 we will construct, under different additional set theoretical assumptions, additive almost continuous (in sense of Stallings) functions ƒ : ℝ → ℝ whose graph is either meager or null in the plane. In Section 3 we will construct an additive almost continuous function ƒ : ℝ → ℝ which has the Cantor intermediate value property but is discontinuous on any perfect set. In particular, such an ƒ does not have the strong Cantor intermediate value property.

In this paper, we give several existence theorems for solutions of the generalized quasi–variational inequalities under new conditions. Even in topological vector spaces, our results are new.

The impulsive equation with retarded argument x ′( t ) + a ( t ) x ( t ) + p ( t ) x ( t – τ) = 0, t ≠ t k , Δ x ( t k ) + a k x ( t k ) + p k x ( t k – τ) = 0, is considered, where the function p ( t ) and the sequence { p k } are not of constant sign. Sufficient conditions are found for oscillation of all solutions to the equation under consideration.

In this paper, we study a class of generalized strongly nonlinear quasivariational inequality problem ( GSNQVIP ( T, A, g, D, K ( x ))) which include the most of quasivariational inequalities and quasicomplementarity problems as special cases. We prove that the generalized strongly nonlinear quasivariational inequality problem is equivalent to solving the set-valued implicit Wiener–Hopf equation. By using the equivalence, a new iterative algorithm for finding the approximate solutions of the generalized strongly nonlinear quasivariational inequality problems are suggested and analyzed. The convergence criteria for the algorithm is also discussed. These new results include many known results for generalized quasivariational inequalities and generalized quasicomplementarity problems as special cases.

The self-gravitating electrodynamic stability of an annular fluid jet (a fluid jet having a tar cylinder as a mantle) pervaded and surrounded by periodic time dependent electric field has been developed. The perturbed system equations describing the model motion is turned to a second integro-differential Mathieu equation. The self-gravitational force is destabilizing only for small axisymmetric perturbation modes. The radii tar-fluid cylinders ratio plays an important role in stabilizing the model. The periodic longitudinal electric field is strongly destabilizing for all perturbation modes. However under some restrictions (independent of the electric field amplitude), it is found that the electric field frequency has a stabilizing influence and that influence suppresses almost the instability character of the annular jet. In contrast to the same model pervaded by classical (constant) electric field, the self-gravitational instability will never be suppressed whatever is the strength of the pervaded electric field. The present analyses have been performed on the basis of the Lagrangian energy principle which it was not an easy job.

We prove an existence and uniqueness for the Einstein–Vlasov system locally in time. Considerations are restricted to initial data which give a solution in harmonic coordinates.

We prove an existence and uniqueness of solutions for the Einstein–Boltzmann system locally in time. We restrict our attention to initial data which gives a solution in harmonic coordinates.