1 Basic Theorems of Complex Analysis 1 1.1 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Infinite Sequences of Complex Numbers . . . . . . . . . . . . . 3 1.3 The Least Upper Bound Principle . . . . . . . . . . . . . . . . 5 1.4 Monotonic Sequences . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Upper and Lower Limits of Bounded Sequences of Real Numbers 7 1.6 The ...

The present paper deals with the study of the notion of an atom of a function m defined on an effect algebra L with values in [0,∞]; a few examples of atoms for null-additive as well as for non-null-additive functions are also given. We have proved a Saks type decomposition theorem for an element a with m(a) > 0 (for a suitable m), which does not contain any atom of m, in a σ-complete effect al...

A field extension R of the real numbers is presented. It has similar algebraic properties as ; for example, all roots of positive numbers exist, and the structure C obtained by adjoining the imaginary unit is algebraically complete. The set can be totally ordered and contains infinitely small and infinitely large quantities. Under the topology induced by the ordering, the set becomes Cauchy co...

This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of...

We consider the helical reduction of the wave equation with an arbitrary source on (n+1)-dimensional Minkowski space, n ≥ 2. The reduced equation is of mixed elliptic-hyperbolic type on Rn. We obtain a uniqueness theorem for solutions on a domain consisting of an n-dimensional ball B centered on the reduction of the axis of helical symmetry and satisfying ingoing or outgoing Sommerfeld conditio...

In this work, we formally proved Descartes Rule of Signs, which relates the number of positive real roots of a polynomial with the number of sign changes in its coefficient list. Our proof follows the simple inductive proof given by Arthan [1], which was also used by John Harrison in his HOL Light formalisation. We proved most of the lemmas for arbitrary linearly-ordered integrity domains (e.g....

f(x) < 0, f(x) > 0. (1) Since f is continuous, the intermediate-value theorem guarantees the existence of some x∗ ∈ I satisfying f(x∗) = 0. The calculation of (1) can be verified by a computer. Hence, (1) describes an existence test for n = 1. We want to point out that the existence tests shall be performed by a computer; i.e., the test (1) shall be done by a computer. But as we will see in the...

Diana Gordon ([email protected])* Devika Subramanian ([email protected])** Mary Haught ([email protected])* Robynn Kobayashi ([email protected])* Sandra Marshall ([email protected])*** *Naval Research Laboratory, Code 5510, Washington, D.C. 20375 **Department of Computer Science, Rice University, Houston TX 77005 ***Department of Psychology, San Diego State Univers...