Volume 31, Issue 3 -

In 1907, E. Study gave a geometric proof of the classical Schwarz reflection principle for harmonic functions. We show that this method can also be used to obtain the more general point-to-point reflection formula for polyharmonic functions in ℝ 2 . The advantage of this method over others is that it avoids use of Garabedian´s generalized Green´s functions.

We present expressions in terms of a double infinite series for the Stieltjes constants γ k (a) . These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case γ k (1) =  γ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function.

In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimensions greater than two. As an application we include an a priori estimate for the corresponding torsion coefficients in arbitrary codimensions.

This paper is concerned with traveling wave solutions to a malignant tumor invasion model. In 1999, Perumpanani et al. presented a mathematical model of malignant tumor cell invasion into connective tissue without cell diffusion. In their and subsequent researches, traveling waves have been studied mainly by numerical methods. In this paper, the existence of smooth traveling wave solutions to the model is proved rigorously. In order to show the existence of traveling waves, one considers heteroclinic orbits on a phase plane and constructs an invariant region for the orbits. Moreover, their asymptotic behavior is investigated. Finally, a remark on discontinuous traveling waves is given.

The paper is about calculating single integrals of the form ∫ 0 1 x m {1/ x } k d x and double fractional part integrals of the form ∫ 0 1 ∫ 0 1 { x / y } m { y / x } k d x d y , where m , k , are nonnegative integers. In particular, we show that the symmetric integral ∫ 0 1 ∫ 0 1 { x / y } m { y / x } m d x d y , equals a rational linear combination of 1 and the zeta numbers ζ ( j ) , j  = 2,⋯, m +1 . We also obtain, as a particular case of our results, new integral representations for the Euler–Mascheroni constant in terms of single and double symmetric fractional part integrals.