A note on machine method for root extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

Heron’s cubic root iteration method with a new approach

Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

The asymptotic formula on average weighted path length for scale-free modular network

In this paper, we discuss the average path length for a class of scale-free modular networks with deterministic growth. To facilitate the analysis, we define the sum of distances from all nodes to the nearest hub nodes and the nearest peripheral nodes. For the unweighted network, we find that whether the scale-free modular network is single-hub or multiple-hub, the average path length grows logarithmically with the increase of nodes number. For the weighted network, we deduce that when the network iteration [Formula: see text] tends to infinity, the average weighted shortest path length is bounded, and the result is independent of the connection method of network.

Remarks on Heron's cubic root iteration formula

The existence as well as the computation of roots appears in number theory, algebra, numerical analysis and other areas. The present study illustrate the contributions of several authors towards the extraction of different order roots of real number. Different methods with several approaches are studied to extract the roots of real number. Some of the methods described earlier are equivalent as observed in the present study. Heron developed a general iteration formula to determine the cube root of a real number N i.e. $\displaystyle\sqrt[3]{N}=a+\frac{bd}{bd+aD}(b-a)$, where $a^3<N<b^3$, $d=N-a^3$ and $D=b^3-N$ . Although the direct proof of the above method is not available in literature, some authors have proved the same with the help of conjectures. In the present investigation, the proof of Heron's method is explained and is generalized for any odd order roots. Thereafter it is observed that Heron's method is a particular case of the generalized method.

Maslov (P, ω)-Index Theory for Symplectic Paths

In this paper, the Maslov (P, ω)-index theory for a symplectic path is developed and the Bott-type iteration formula is proved.

EQUIVARIANT SYMPLECTIC HOMOLOGY AND MULTIPLE CLOSED REEB ORBITS

We study the existence of multiple closed Reeb orbits on some contact manifolds by means of S1-equivariant symplectic homology and the index iteration formula. We prove that a certain class of contact manifolds which admits displaceable exact contact embeddings, a certain class of prequantization bundles, and Brieskorn spheres have multiple closed Reeb orbits.