One of the remarkable things about this theorem is the way in which it suggests that geometry informs arithmetic. The geometric genus g is a manifestly geometric condition, yet it is controlling what seems to be an arithmetic property. Why should the number of integral solutions to xn + yn = zn have anything to do with the shape of the complex solutions? You might argue that that the genus is e...

We prove a result which implies that, for any real numbers a and b satisfying 0 ≤ a ≤ b ≤ 1, there exists an infinite sequence of positive integers A with lower density a and upper density b such that the sets A and N \ A contain no infinite arithmetic and geometric progressions. Furthermore, for any m ≥ 2 and any positive numbers a1, . . . , am satisfying a1 + · · · + am = 1, we give an explic...

Mixed arithmetic and geometric means, with and without weights, are both considered. Related to mixed arithmetic and geometric means, the following three types of inequalities and their generalizations, from three variables to a general n variables, are studied. For arbitrary x, y, z ≥ 0 we have [ x + y + z 3 (xyz) ]1/2 ≤ ( x + y 2 · y + z 2 · z + x 2 )1/3 , (A) 1 3 (√ xy + √ yz + √ zx ) ≤ 1 2 ...

We present a completeness proof of the inductive assertion method for object-oriented programs extended with auxiliary variables. The class of programs considered are assumed to compute over structures which include the standard interpretation of Presburger arithmetic. Further, the assertion language is first-order, i.e., quantification only ranges over basic types like that of the natural numb...

in this study, some seed physical properties of two sugar beet varieties, shirin and gadouk (436), were evaluated as a function of moisture content. physical properties included length, width, thickness, arithmetic and geometric mean diameters, sphericity, thousand seed weight, angle of repose, terminal velocity, true density, bulk density, porosity, and coefficient of friction. four moisture c...