This talk should be regarded as an elementary introduction to differential algebra. It culminates in a purely algebraic proof, due to M. Rosenlicht [Ros2], of an 1835 theorem of Liouville on the existence of “elementary” integrals of “elementary” functions. The precise meaning of elementary will be specified. As an application of that theorem we prove that the indefinite integral ∫ e 2 dx canno...

Reading a book is also kind of better solution when you have no enough money or time to get your own adventure. This is one of the reasons we show the ia 64 and elementary functions speed and precision as your friend in spending the time. For more representative collections, this book not only offers it's strategically book resource. It can be a good friend, really good friend with much knowledge.

We show that if M is a stable unsuperstable homogeneous structure, then for most κ < |M| , the number of elementary submodels of M of power κ is 2 . Through out this paper we assume that M is a stable unsuperstable homogeneous model such that |M| is strongly inaccessible (= regular and strong limit). We can drop this last assumption if instead of all elementary submodels of M we study only suit...

Let M(t) denote the time required to multiply two t-digit numbers using base b arithmetic. Methods are presented for computing the elementary functions in O( t 1/3M(t) ) time.

The SHUNYATA program constructs proof methods by analyzing proofs of simple theorems in mathematical theories such as group theory and uses these methods to form prooh of new theorems in the same or in other theories. Such methods are capable of generating proo& of theorems whose complexity represents the state of the art in automated theorem proving. They are composed of elementary functions s...

We consider a special class of two-dimensional discrete equations defined by relations on elementary N ×N squares, N > 2, of the square lattice Z, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary N × N squares, N > 2, in the cubic lattice Z. For an arbitrary N we prove such consistency on cubic lattices for tw...

There is a general method, applicable in many situations, whereby a pseudo–Riemannian metric, invariant under the action of some Lie group, can be deformed to obtain a new metric whose geodesics can be expressed in terms of the geodesics of the old metric and the action of the Lie group. This method applied to Euclidean space and the unit sphere produces new examples of complete Riemannian metr...

We propose to apply Simplicity Theory (ST) to model interest in creative situations. ST has been designed to describe and predict interest in communication. Here we use ST to derive a decision rule that we apply to a simplified version of a creative game, the Poietic Generator. The decision rule produces what can be regarded as an elementary form of creativity. This study is meant as a proof of...