Hobson's formula for Dunkl operators and its applications

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.     

Case study projects for college mathematics courses based on a particular function of two variables

Based on a sequence of number pairs, a recent paper (Mauch, E. and Shi, Y., 2005, Using a sequence of number pairs as an example in teaching mathematics, Mathematics and Computer Education, 39(3), 198–205) presented some interesting examples that can be used in teaching high school and college mathematics classes such as algebra, geometry, calculus, and linear algebra. In this paper, this study is generalized further to develop a few interesting case study proposals that can be used for student projects in college mathematics courses such as real functions, analytic geometry, and complex variables. In addition to using them in individual courses, these studies may also be combined to offer seminars or workshops to college mathematics students. Projects like these are likely to promote student interest and get students more involved in the learning process, and therefore make the learning process more effective.     

On Hill's operator with a matrix potential

In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the self‐adjoint operator generated by a system of Sturm–Liouville equations with summable coefficients and quasiperiodic boundary conditions. Then using these asymptotic formulas, we find conditions on the potential for which the number of gaps in the spectrum of the Hill's operator with matrix potential is finite. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability criteria for differential equations with variable time delays

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results.     

Opial's and wirtinger's type discrete inequalities in two independent variables

We develop discrete Taylor's formula in two independent variables and use it to establish Opial's type inequality involving higher order partial differences. This is followed by several variations of discrete Wirtinger's inequality in two independent variables     

On the lower bound for a class of harmonic functions in the half space

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n ≥ 2. To this end, we first generalize the Carleman's formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna's representation for harmonic functions in the half sphere by using Hörmander's theorem.     

Integral representation and asymptotic behavior of harmonic functions in half space

Using Carleman's formula of a harmonic function in the half space and Nevanlinna's representation of a harmonic function in the half sphere, we prove that a harmonic function, whose positive part satisfies a slowly growing condition, can be represented by a certain integral. This improves some classical Poisson integrals for harmonic functions.     

The Pythagorean theorem and the Euclidean parallel postulate

In this note, we develop new, simple and very accurate asymptotic approximations for non-integer order derivatives of monomial functions by using the more accurate asymptotic approximations for large factorials that have recently appeared in the literature.     

相对于环的自同态的(b, c)-逆

研究了相对于环的自同态的$(b, c)$-逆的相对性质.在更一般地条件下, 定义了一类新的广义逆, 即$\alpha$-$(b, c)$-逆. 通过具体例子说明了$\alpha$-$(b, c)$-逆与$(b, c)$-逆是完全不同的两类广义逆, 并研究了$\alpha$-$(b, c)$-逆与$(b, c)$-逆等价的条件. 此外, 还研究了$\alpha$-$(b, c)$-逆的强clean分解. 这些研究结果统一和推广了$(b, c)$- 逆上的若干已知结果.     

Recursive Structures and Ershov's Hierarchy

Ash and Nerode [2] gave natural definability conditions under which a relation is intrinsically r. e. Here we generalize this to arbitrary levels in Ershov's hierarchy of Δ sets, giving conditions under which a relation is intrinsically α-r. e. Mathematics Subject Classification: 03C57, 03D55.     

On a reverse Hardy-Littlewood-Polya's inequality

By the use of the weight coefficients, the idea of introduced parameters and Euler-Maclaurin summation formula, a reverse Hardy-Littlewood-Polya''s inequality with parameters as well as the equivalent forms are provided. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are given.     

Weighted Hardy's inequality in a limiting case and the perturbed Kolmogorov equation

In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant on the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein–Uhlenbeck operator perturbed by a critical singular potential in a two-dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the limiting case of results which are established in the work of Goldstein et al. [Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential. Appl Anal. 2012;91(11):2057–2071] and Hauer and Rhandi [A weighted Hardy inequality and nonexistence of positive solutions. Arch Math. 2013;100:273–287] in the non-critical cases, and are also considered as extensions of a result in Cabré and Martel [Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potential singulier. C R Acad Sci Paris Sér I Math. 1999;329:973–978] to the Kolmogorov operator case perturbed by a critical singular potential.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.