This paper explores the role of group theory in providing a proof for the Law of Quadratic Reciprocity, which states that for distinct odd primes pand q, q is a quadratic residue mod p if and only if p is a quadratic residue mod q, unless p and q are both congruent to 3 mod 4. The Law of Quadratic Reciprocity is an important result in number theory; it provides us with a simple method to determ...

Harmonic Distortion in many of the industrial applications are occur primarily owing to enormous utilization loads with high non-linearity like power converters, speed varying drives and arc furnaces. The semiconductor is used achieve variation conversion from one source another. Mostly active filters tuned utilized remove harmonic included current. passive inductance inserted line reduces harm...

1 CTL formulae We formalize basic concepts of Computational Tree Logic (CTL) [2, 1] within the simply-typed set theory of HOL. By using the common technique of “shallow embedding”, a CTL formula is identified with the corresponding set of states where it holds. Consequently, CTL operations such as negation, conjunction, disjunction simply become complement, intersection, union of sets. We only ...

In 1927, the topologist Karl Menger published an article called Zur allgemeinen Kurventheorie (On the general theory of curves), in which he stated a remarkable result, now one of the most fundamental results in graph theory: Satz. Ist K ein kompakter regull ar eindimensionaler Raum, welcher zwi-schen den beiden endlichen Mengen P und Q n-punktig zusammenhh angend ist, dann enthh alt K n paarwe...

We give a method to embed the q-series in a (p, q)-series and derive the corresponding (p, q)-extensions of the known q-identities. The (p, q)-hypergeometric series, or twin-basic hypergeometric series (different from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]p,q = (p n−qn)/(p−q). This twin-basic number occurs in the theory of two-parameter quantu...

It was recently shown by Witten on the basis of several examples that the topological B-model whose target space is a Calabi-Yau (CY) supermanifold is equivalent to holomorphic Chern-Simons (hCS) theory on the same supermanifold. Moreover, for the supertwistor space CP 3|4 as target space, it has been demonstrated that hCS theory onCP 3|4 is equivalent to self-dual N = 4 super Yang-Mills (SYM) ...

Selection of proper reference compensation current extraction scheme plays the most crucial role in the performance of an active power filter (APF). This paper mainly describes three different control schemes used in APFs namely, Conventional instantaneous active and reactive power (p–q), Modified p–q, and Instantaneous active and reactive current component (id–iq) schemes. Our objective here i...