Volume 11, Issue 2

We prove that the Covering Property Axiom , which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function ƒ : ℝ → ℝ whose graph is of measure zero. We also show that, under , there exists a Hamel basis H for which, E + ( H ), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin's axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that implies its seemingly stronger version, in which ω 1 -many games are played simultaneously.

New existence results are presented for fuzzy differential and integral equations. Our analysis combines the stacking theorem with results concerning the maximal solution for an appropriate differential equation.

Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable subspaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI ( P α : ω ≤ α ≤ λ) such that any projection is the sum of an operator in the closure of the linear span of countably many P α 's (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprāns' construction provides an unconditional example for λ = ω 1 . We note that it is impossible for λ > ω 2 .

Let C ( X ; ℝ) the algebra of continuous real valued functions defined on a locally compact space X . We consider linear subspaces 𝔸 ⊂ C ( X ; ℝ) having the following property: there is a sequence (Φ j ) j ∈ℕ of positive functions in 𝔸 with lim x →∞ Φ j ( x ) = +∞ for every j ∈ ℕ, such that 𝔸 consists of functions ƒ ∈ C ( X ; ℝ) bounded above for the absolute value by an homothetic of some Φ n ( n depends on each ƒ). Dominated convergence of a sequence ( g n ) n ≥1 in 𝔸 is an estimation of the form | g n ( x ) – g ( x )| ≤ ε n | h ( x )| for all x ∈ X and all n ∈ ℕ where g n , g, h ∈ 𝔸 and ε n → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices 𝔹 ⊂ 𝔸 and we show that the dominated convergence for sequences is exactly the convergence of sequences when 𝔸 is endowed with a locally convex ( DF )-space topology.

Fritz John and Kuhn-Tucker type necessary optimality conditions for a Pareto optimal (efficient) solution of a multiobjective control problem are obtained by first reducing the multiobjective control problem to a system of single objective control problems, and then using already established optimality conditions. As an application of Kuhn-Tucker type optimality conditions, Wolfe and Mond-Weir type dual multiobjective control problems are formulated and usual duality results are established under invexity/generalized invexity, relating properly efficient solutions of the primal and dual problems. Wolfe and Mond-Weir type dual multiobjective control problems with free boundary conditions are also presented.

Let ( X, T X ) be a topological space and let ( Y, d Y ) be a metric space. For a function ƒ : X → Y denote by C (ƒ) the set of all continuity points of ƒ and by D (ƒ) = X \ C (ƒ) the set of all discontinuity points of ƒ. Let C ( X, Y ) = {ƒ : X → Y ; ƒ is continuous}, H ( X, Y ) = {ƒ : X → Y ; D (ƒ) is countable}, H 1 ( X, Y ) = {ƒ : X → Y ; ∃ h ∈ C ( X, Y ) { x ; ƒ( x ) ≠ h ( x )} is countable}, and H 2 ( X, Y ) = H ( X, Y )∩ H 1 ( X, Y ). In this article we investigate some convergences (pointwise, uniform, quasiuniform, discrete and transfinite) of sequences of functions from H ( X, Y ), H 1 ( X, Y ) and H 2 ( X, Y ).

This paper describes a wide class of coupled KdV equations. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2-component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 metric on the semidirect product group , where Diff( S 1 ) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg-Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on .

The present paper is focused on the analysis of three very simple models of carcinogenesis mutations that are based on reaction-diffusion systems and Lotka-Volterra food chains. We consider the case with two stages of mutations and study the systems of three reaction-diffusion equations with zero-flux boundary conditions. We focus on the Turing instability and show that this type of instability is not possible for these models. We also propose some modifications of the considered equations. Results are illustrated by computer simulations.