The notion that science is unified in one way or another dates back at least to Aristotle, though unity claims since then have been diverse and variously motivated. By way of introduction to the modern discussion of unity, disunity, and integration, in this first section we examine five historical attempts to unify knowledge: Aristotle’s metaphysical and hierarchical unity; the Enlightenment pr...

Let p ≡ 1 (mod 4) be a prime number and let ζ = e be a primitive root of unity. Then there exists a unique biquadratic extension eld Q(y)/Q that is a sub eld of Q(ζ). The aim of this work is to construct an algorithm for nding such y explicitly. Finally we state a general conjecture about the y we found.

Recently, the authors introduced two new second-order single-resistance-controlled oscillators (SRCO) employing four/three unity-gain cells (unity-gain voltage followers and unity-gain current followers). In this communication, a novel family of SRCOs has been presented which employ a reduced number of (only two) unity-gain cells as active elements. The workability of the new circuits has been ...

The partition of unity is an essential ingredient for meshless methods named by GFEM, PUFEM (partition of unity FEM), XFEM(extended FEM), RKPM(reproducing kernel particle method), RPPM(reproducing polynomial particle method), the method of clouds in the literature. There are two popular choices for partition of unity: a piecewise linear FEM mesh and the Shepard-type partition of unity. However,...

7 Periodic Series and Functions 128 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 128 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 130 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 131 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 133 7.5 The Exponential of a Complex Number . . . . . . . . . . . . . 133 7.6...

In this paper, we show that the problem of computing the complex roots of unity is not as simple as it seems at rst. In particular, the formulas given in a standard programmer's reference book Knuth, Seminumerical Algorithms, 1981] are shown to be numerically unstable , giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both ...

Let L/K be an abelian extension of number fields. We prove an uniform lower bound for the height in L∗ outside roots of unity. This lower bound depends only on the degree [L : K].

There exists a number of known constructions for perfect sequences over roots of unity. These include Frank sequences of length n over n roots of unity [Heimiller, 1961][Frank, 1962], Chu sequences of length n over n roots of unity for n odd and length n over 2n roots of unity for n even [Chu, 1972], Milewski sequences of length m overm roots of unity [Milewski, 1983], Liu-Fan sequences of leng...

We construct an invariant JM of integral homology spheres M with values in a completion Ẑ[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten-ReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhin-Turaev invariants of M . As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) a...