From natural numbers to numbers and curves in nature – I

The interconnection between number theory, algebra, geometry and calculus is shown through Fibonacci sequence, golden section and logarithmic spiral. In this two-part article, we discuss how simple growth models based on these entities may be used to explain numbers and curves abundantly found in nature

From natural numbers to numbers and curves in nature - II

In this second part of the article we discuss how simple growth models based on Fibbonachi numbers, golden section, logarithmic spirals, etc. can explain frequently occuring numbers and curves in living objects. Such mathematical modelling techniques are becoming quite popular in the study of pattern formation in nature

Optimization problems in elementary geometry

Optimization, a principle of nature and engineering design, in real life problems is normally achieved by using numerical methods. In this article we concentrate on some optimization problems in elementary geometry and Newtonian mechanics. These include Heron's problem, Fermat's principle, Brachistochrone problems, Fagano's problem, geodesics on the surface of a parallelepiped, Fermat/Steiner problem, Kakeya problem and the isoperimetric problem. Some of these are very old and historically famous problems, a few of which are still unresolved. Close connection between Euclidean geometry and Newtonian mechanics is revealed by the methods used to solve some of these problems. Examples are included to show how some problems of analysis or algebra can be solved by using the results of these geometrical optimization problems

Curious consequences of simple sequences

Simple sequences and series of natural numbers, their reciprocals, integer powers of natural numbers and their reciprocals are considered. Some interesting physical and mathematical consequences of these are discussed

Magnetic anisotropy, first-order-like metamagnetic transitions and large negative magnetoresistance in the single crystal of Gd2_{2}PdSi3_3

Electrical resistivity ($\rho$), magnetoresistance (MR), magnetization, thermopower and Hall effect measurements on the single crystal Gd$_{2}$PdSi$_3$, crystallizing in an AlB$_2$-derived hexagonal structure are reported. The well-defined minimum in $\rho$ at a temperature above N\'eel temperature (T$_N$= 21 K) and large negative MR below $\sim$ 3T$_N$, reported earlier for the polycrystals, are reproducible even in single crystals. Such features are generally uncharacteristic of Gd alloys. In addition, we also found interesting features in other data, e.g., two-step first-order-like metamagnetic transitions for the magnetic field along [0001] direction. The alloy exhibits anisotropy in all these properties, though Gd is a S-state ion.Comment: RevTeX, 5 pages, 6 encapsulated postscript figures; scheduled to be published in Phy. Rev. B (01 November 1999, B1

Parametrically excited Non-linear traveling beams with and without external forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations

Non-linear vibration of a travelling beam having an intermediate guide

The free and forced responses of a non-linear travelling beam having an intermediate guide have been analysed. The guide is modelled as a purely elastic constraint with no inertia. While a suitably located guide increases the natural frequencies, the friction present in the guide-beam interface tends to destabilize the system. The presence of the guide reduces the vibration level by avoiding resonance conditions. The effect of the non-linear term is very sensitive to the location of the guide if the guide stiffness is small. It is suggested that the guide is placed near the antinode of the predominantly excited mode

Engineering Associated with Research and Development

It is found, at the beginning of the civilisation, technology was developed without science. In later stages technology followed science . At present both run parallely. Though engineering plays a role in basic research , it is more dominant in applied research.Research normally becomes more fruitful when its results are translated to the benefit of the society through technology development and transfer. The R&D starts with conception of an innovative idea and followed by evaluation, accomplishment, development and culminatein release. The R&D scientist see their ideas to reality through three distinct parties - management, customer and engineering. The engineering aspects of R&D can be generalised through four phases viz., planing and specification, design, process/prototype and release. Earlier it was thought it is better not to control R&D. When competitiveness increased,control became necessary. Attempt is made to enumerate the role of engineering in R &D in nutshell in this paper. It is also demonstrated that effective innovation management synergising, the efforts of engineering and R&D teams can enhance the quality of collaborative R&D and promotes cutting edge over competition from multinationals. This would also ensure greater confidence and faith in the domestic industry towards indigenous stand-alone R&D institutions