Volume 23, Issue 2

Let I=[0,1]{I=[0,1]}, and let b⁢B1{bB_{1}} be the set of Baire-1 self-maps of I . For f∈b⁢B1{f\in bB_{1}}, let Λ⁢(f)=⋃x∈Iω⁢(x,f){\Lambda(f)=\bigcup_{x\in I}\omega(x,f)} be the set of ω-limit points of f . We prove the following: (i) There exists a residual subset S of b⁢B1{bB_{1}} such that for any f∈S{f\in S} and x∈I{x\in I} the ω-limit set ω⁢(x,f){\omega(x,f)} is contained in the set of points at which f is continuous, and ω⁢(x,f){\omega(x,f)} is an ∞{\infty}-adic adding machine. (ii) There exists a residual subset S of b⁢B1{bB_{1}} such that for any f∈S{f\in S} and for any ε>0{\varepsilon>0} there exists a natural number M such that fm⁢(I)⊂Bε⁢(Λ⁢(f)){f^{m}(I)\subset B_{\varepsilon}(\Lambda(f))} whenever m>M{m>M}. Moreover, f:Λ⁢(f)→Λ⁢(f){f:\Lambda(f)\rightarrow\Lambda(f)} is a bijection.

A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the special homogeneous spaces are derived by using the general quotient integral formula. Finally, our results are supported by some examples.

Using the results in [12] where a construction of the Dunkl intertwining operator for a large set of regular parameter functions is provided, we establish an integral expression for the Dunkl kernel in the context of the dihedral group 𝒟n{\mathcal{D}_{n}} with constant parameter function k∈ℂ{k\in\mathbb{C}} and arbitrary order n≥2{n\geq 2}. Our main tool is a differential system that leads to the explicit expression of the Dunkl kernel whenever an appropriate solution of it is obtained. In particular, an explicit expression of the Dunkl kernel Ek⁢(x,y){E_{k}(x,y)} is given when one of its argument x or y is invariant under the action of any reflection in the dihedral group. We obtain also a generating series for the homogeneous components Km⁢(x,y){K_{m}(x,y)}, m∈ℤ+{m\in\mathbb{Z}^{+}}, of the Dunkl kernel and provide new sharp estimates for the Dunkl kernel in the large context k∈ℂ{k\in\mathbb{C}}, n≥2{n\geq 2} and -2⁢n⁢k≠1,2,3,…{-2nk\neq 1,2,3,\dots\,}.

Starting with some equations that characterize the Takagi function in the class of bounded functions, we come naturally to a family of functional equations and extend two classical results by H.-H. Kairies to that family.

We investigate the stability features of steady-states of a two-dimensional system of ferromagnetic nanowires. We constitute a system with the finite number of nanowires arranged on the (e→1,e→2){(\vec{e}_{1},\vec{e}_{2})} plane, where (e→1,e→2,e→3){(\vec{e}_{1},\vec{e}_{2},\vec{e}_{3})} is the canonical basis of ℝ3{\mathbb{R}^{3}}. We consider two cases: in the first case, each nanowire is considered to be of infinite length, whereas in the second case, we deal with finite length nanowires to design the system. In both cases, we establish a sufficient condition under which these steady-states are shown to be exponentially stable.

We consider the problem of existence and uniqueness of extensions for the solutions of the Pexider type equation f⁢(x+y)=g⁢(x)+h⁢(y)+k⁢(x)⁢l⁢(y){f(x+y)=g(x)+h(y)+k(x)l(y)} for (x,y)∈D{(x,y)\in D}, where X is a normed space and D is a nonempty open and connected subset of X2{X^{2}}.

A higher order difference equation is studied. The equation is defined on ℤ{{\mathbb{Z}}} and contains a p -Laplacian and both advance and retardation. Some criteria are established for the existence of infinitely many anti-periodic solutions of the equation. Several consequences of the main theorems are also included. Two examples are provided to illustrate the applicability of the results.

Interest rate risk for portfolios with random cash flows is mitigated via minimizing its ℒ2{\mathcal{L}^{2}}-measure. A link between monotonicity of linear operators and immunization under cone restrictions is examined. The results are based on the Hilbert space methods applied to incomplete markets.

We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure p appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint div⁡u=0{\operatorname{div}u=0}.

This work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.