BACKGROUND AND PURPOSE The size of vestibular aqueducts (VAs) seen on CT studies varies. The current practice of calling a VA enlarged when it exceeds a certain threshold (eg, 1.5 mm at the midpoint) is arbitrary. Our hypothesis was that statistical analysis of the range of VA widths in a normal-hearing population would lead to a mathematic definition of the upper-limit-of-normal VA width. MA...

It is a striking fact that differential calculus exists not only in analysis (based on the real numbers R and limits therein), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the “affine line” R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if xk = 0 for suit...

Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for inquiry, number of extensions midpoint estimations are studied. The goal this research article is discover novel midpoint-type that twice q?2-differentiable (?,m)-convex functions. Firstly, we obtain identity q?2-integral by employing calc...

In this paper‎, ‎making use of a new identity‎, ‎we establish new‎ ‎inequalities of Ostrowski type for the class of preinvex functions and‎ ‎gave some midpoint type inequalities‎.