Volume 28, Issue 5

This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type - div ⁡ ( a ⁢ ( x , u , ∇ ⁡ u ) ) + H ⁢ ( x , u , ∇ ⁡ u ) = f {-\operatorname{div}(a(x,u,\nabla u))+H(x,u,\nabla u)=f} .

Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\mathfrak{R}} . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

In this paper, we use some methods of combinatorial set theory, in particular, the ones related to the construction of independent families of sets and also some modified version of the notion of small sets originally introduced by Riečan and Neubrunn, to give an abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby related to a nonseparable invariant extension of the Lebesgue measure in spaces with transformation groups.

The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

We prove the following theorem. Let α = a + i ⁢ b {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either b ≠ 0 {b\neq 0} or b = 0 {b=0} and a > 0 {a>0} . Let ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} be independent complex random variables. Assume that the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent. Then ξ j {\xi_{j}} are degenerate random variables. (ii) Let b = 0 {b=0} and a < 0 {a<0} . Then there exist complex Gaussian random variables in the wide sense ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent.

Oscillation criteria for a class of second-order delay difference equations with a superlinear neutral term are established using a new approach. The results improve and significantly simplify the ones reported in the literature.

In the paper, some properties of functions continuous with respect to a density type strong generalized topology are presented. In particular, it is proved that each real function is approximately continuous with respect to this generalized topology almost everywhere. Moreover, some separation axioms for this generalized topological space are investigated.

We establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality ∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n ) \lVert I_{\alpha}f\rVert_{L^{p}_{v}(\mathbb{R}^{n})}\leq C\lVert f\rVert_{L^{p% ,1}(\mathbb{R}^{n})} for the Riesz potential I α {I_{\alpha}} , where L p , 1 ⁢ ( ℝ n ) {L^{p,1}(\mathbb{R}^{n})} is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

The purpose of this paper is to give a degree of approximation of a function in the space H p ω {H^{\omega}_{p}} with norm ∥ ⋅ ∥ p ω {\lVert\,\cdot\,\rVert^{\omega}_{p}} by using even-type delayed arithmetic mean of its Fourier series.

In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

The purpose of this paper is to provide a cohomology of n -Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

In this paper, we define consistent criteria for hypotheses testing, such that the probability of any kind of error is zero for a given criterion. The necessary and sufficient conditions for the existence of such criteria are given.

The quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of q -special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the q -Tricomi functions and 2D q -Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the q -Tricomi and q -Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the q -Tricomi and 2D q -Tricomi functions in the context of the irreducible representation ( ω ) {(\omega)} of the two-dimensional quantum algebra ℰ q ⁢ ( 2 ) {\mathcal{E}_{q}(2)} is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter q -Tricomi functions and their relationship with the 2-variable 1-parameter q -Bessel functions are also explored.

The support of a wavelet transform associated with a square integrable irreducible representation of a homogeneous space is shown to have infinite measure. Assumptions are illustrated and supported by examples. The pointwise homogeneous approximation property for a wavelet transform has been investigated. An analogue of Heisenberg type inequality is also obtained for a wavelet transform on a wavelet group.

In this paper, for a discrete hypergroup K and its subhypergroup H , we initiate the related Hecke * {*} -algebra which is an extension of the classical one, and study its basic properties. Especially, we give a necessary and sufficient condition (named (β)) for this algebra to be associative. Also, we show that this new structure is an associative * {*} -algebra if and only if K is a locally compact group.