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Oman is located within Arid and Semi-Arid regions, where water resources are very scarce. The national average rainfall is just 80-100 mm per year. With high temperatures (30 ºC to above 40 ºC) the evaporation is also high. Therefore, it is absolutely necessary to conserve and recycle water in Oman. The study aims to identify ablution water (the ritual washing of the exposed parts of the body, the face, hands and feet before each prayer called ablution) and to prove that it as one of the least polluted sources of greywater, by examining the properties of greywater and comparing its results with the results of other greywater, taking into account the criteria for reusing treated water, as well as developing appropriate treatment techniques. The study dealt with the use of solar energy to operate the treatment unit, and developed an appropriate design for the treatment unit. The study recommends using the total volume of greywater generated for non-potable purposes including rinsing, car washing, gardening and irrigation purposes. In this way, the greywater from mosques is recycled in the most beneficial way and at a lower cost in order to provide fresh water for future generations.

The world-wide energy demand is continuously increasing compounded with global warming. There is an ever-increasing demand in energy accompanied with depletion of fuel stocks, particularly the fossil fuels. The current energy consumption rates are unsustainable and thus have prompted for an active exploration and use of renewable energy sources. In this paper, we shall explore the future of the renewable energy in Oman based on the case study of Salalah city located in Southern Oman. The electricity shortages and the challenges to overcome the increase in electrical demands for the near future are discussed in detail. The climate of any locality is predominantly determined by the amount of sunshine it receives. The duration of sunshine was measured every one hour using the routine sunshine recorders. Our investigations suggest that solar and wind energy will play an important role in the future of renewable energy in Oman. The paper finishes with some conclusions and recommendations. Based on the basic information of Oman solar radiation data, the application of solar energy has a high opportunity to support Oman’s power generating capacity. Keywords: Renewable energy, Solar Energy, Wind Energy, Oman. JEL Classifications: Q2, Q3, Q4DOI: https://doi.org/10.32479/ijeep.11855

The main aim of this article is to start with an expository introduction to the trigonometric ratios and then proceed to the latest results in the field. Historically, the exact ratios were obtained using geometric constructions. The geometric methods have their own limitations arising from certain theorems. In view of the certain limitations of the geometric methods, we shall focus on the powerful techniques of equations in deriving the exact trigonometric ratios using surds. The cubic and higher-order equations naturally arise while deriving the exact trigonometric ratios. These equations are best expressed using the expansions of the cosines and sine of multiple angles using the Chebyshev polynomials of the first and second kind respectively. So, we briefly present the essential properties of the Chebyshev polynomials. The equations lead to the question of reduced polynomials. This question of the reduced polynomials is addressed using the Euler's totient function. So, we describe the techniques from theory of equations and reduced polynomials. The trigonometric ratios of certain rational angles (when measured in degrees) give rise to rational trigonometric ratios. We shall discuss these along with the related theorems. This is a frontline area of research connecting trigonometry and number theory. Results from number theory and theory of equations are presented wherever required.

An introduction to irrational numbers is done using basic trigonometry, generally taught in pre-university courses. The irrational sets of trigonometric ratios of rational angles are discussed in detail along with two major theorems and their proofs. Algebraic numbers along with transcendental numbers are also covered.

The Tsallis $q$-exponential function $e_q(x) = (1+(1-q)x)^{\frac{1}{1-q}}$ is found to be associated with the deformed oscillator defined by the relations $\left[N,a^\dagger\right] = a^\dagger$, $[N,a] = -a$, and $\left[a,a^\dagger\right] = \phi_T(N+1)-\phi_T(N)$, with $\phi_T(N) = N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed oscillator the annihilation operator $a$ corresponds to a deformed derivative with the Tsallis $q$-exponential functions as its eigenfunctions, and the Tsallis $q$-exponential functions become the coherent states of the deformed oscillator. When $q = 2$ these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when $q = 1$ this deformed oscillator is a canonical boson oscillator, when $1 < q < 2$ its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when $q \longrightarrow 2$ it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state., Comment: Third version in which three more references have been added. To appear in International Journal of Theoretical Physics

The scalar optics is based on a Hamiltonian derived using the Fermat's principle of least time. The same Hamiltonian can now be derived from the Maxwell equations. The Helmholtz equations has a striking mathematical similarity to the Klein–Gordon equation for the relativistic spin-0 particle. It is possible to make use of this strong similarity through quantum techniques and develop an alternative to the traditional approaches. The non-traditional formalism of Helmholtz optics reproduces the traditional results as expected. Moreover, it leads to the wavelength-dependent modifications of the paraxial as well as the aberrating behavior. To illustrate the formalism, we consider the propagation of light in optical fibers in substantial detail. We also consider the propagation of light through a graded index slab.

It is possible to apply quantum methodologies to the matrix-representation of Maxwell’s equations. This results in a unified formalism of light beam optics and light polarization, accompanied with wavelength-dependent contributions.

Scalar theory of quantum electron beam optics, at the single-particle level, derived from the Dirac equation using a Foldy-Wouthuysen-like transformation technique is considered. Round magnetic electron lenses with Glaser and power law models for the axial magnetic field $B(z)$ are studied. Paraxial quantum propagator for the Glaser model lens is obtained in terms of the well known fundamental solutions of its paraxial equation of motion. In the case of lenses with the power law model for $B(z)$ the well known fundamental solutions of the paraxial equations, obtained by solving the differential equation, are constructed using the Peano-Baker series also. Quantum mechanics of aberrations is discussed briefly. Role of quantum uncertainties in aberrations, and in the nonlinear part of the equations of motion for a nonparaxial beam, is pointed out. The main purpose of this article is to understand the quantum mechanics of electron beam optics though the influence of quantum effects on the optics of present-day electron beam devices might be negligible., 40 pages. Two equations, (24) and (25), have been added. To appear in Optik

A unified formalism of light beam optics and light polarization is presented. The starting point of our formalism is an exact eight-dimensional matrix-representation of Maxwell's equations in an inhomogeneous medium, which is presented in detail. The beam-optical Hamiltonians are derived without any specifications on the varying refractive index. The new formalism generalizes the traditional and non-traditional treatments of Helmholtz optics. As for the light polarization, the elegant Mukunda–Simon–Sudarshan rule for transition from scalar optics to vector wave optics is obtained as the paraxial limit of the general formalism presented here. The new formalism is a suitable candidate to extend the traditional theory of polarization beyond the paraxial approximation. The unified formalism light beam-optics and light polarization advances the Hamilton's optical-mechanical analogy into the wavelength-dependent regime.

We explore the ways to linearize the wave equations. Special emphasis is paid to the Klein–Gordon equation for a spin-0 relativistic particle and the Helmholtz equation governing scalar optics. Owing to the mathematical similarity, both of these equations are linearized using the Feshbach–Villars procedure. Maxwell's equations are linear but coupled and constrained. So, a matrix representation is presented. New formalisms of beam optics are presented using the linearized form of the Helmholtz equations and the Dirac- like matrix form of Maxwell's equations respectively. It is shown that the matrix formulation of the Maxwell optics naturally leads to a unified treatment of beam optics (including aberrations to all orders) and light polarization, from a single parent Hamiltonian. The non-traditional treatments using quantum methodologies lead to wavelength-dependent modifications of the traditional prescriptions. In the limit of low wavelength, the non-traditional prescriptions of both Helmholtz optics and Maxwell optics presented here reproduce the ‘Lie algebraic formalism of light beam optics’. The accompanying machinery of the Foldy–Wouthuysen transformation technique is described. From the new prescriptions of light beam optics, it is seen that the Hamilton's optical-mechanical analogy persists in the wavelength-dependent regime.

Hamilton's optical–mechanical analogy between the trajectory of material particles in potential fields and the path of light rays in media with continuously variable refractive index played an important role in the development of Schrodinger's wave mechanics. We shall examine how the Hamilton's analogy between charged-particle beam optics and light beam optics is extended to the wavelength-dependent regime. Brief accounts of the quantum prescriptions of charged-particle beam optics along with the non-traditional prescriptions of light beam optics are also presented. These prescriptions have been instrumental in seeing the analogy in the wavelength-dependent regime.

It is well-known that the Helmholtz equation describing scalar optics has a striking mathematical similarity with the Klein–Gordon equation of relativistic quantum mechanics. Exploiting this similarity, quantum methodologies are applied to the Helmholtz equation leading to a new formalism of scalar optics. Paraxial and aberrating Hamiltonians are derived for a general spatially varying refractive index. The beam-optical Hamiltonians thus derived give rise to wavelength-dependent contributions. Example of the graded-index medium is covered in detail. Explicit formulae relating the incident and emergent light rays for the general case of a thick lens are also presented.

The way to generalize scalar to wave optics, thus including polarization in the treatment consistent with the Maxwell equations was shown by Mukunda, Simon and Sudarshan for paraxial systems, based on a group theoretical analysis. Here, the Mukunda–Simon–Sudarshan (MSS) theory for the passage from scalar to vector optics is derived by casting the basic formalism in a framework very similar to the Dirac electron theory. The resulting formalism is suitable for extending the MSS-theory beyond the paraxial approximation.

Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the nonrelativistic Schr\"{o}dinger equation. It is found that the quantum transfer map contains the classical transfer map as the main part and there are tiny quantum correction terms. The negligibly small quantum corrections explain the remarkable success of classical mechanics in charged particle beam optics., Comment: 21 pages

The year 2015 was the United Nations’ International Year of Light and Light-based Technologies. The year long activities created a forum for scientists and engineers and all others inspired by light, to interact with each other and with the public so as to learn more about the nature of light and its many applications. It was also a time to promote and celebrate the Medieval Arab contributions to optics. The year 2015 marked the 1000th anniversary of the seven volume encyclopedia on optics written by the great Arab scientist, Ibn al-Haytham. Here, we present the Medieval Arab achievements in optical sciences and its impact on the European renaissance.

The UN designated International Year of Light and Light-based Technologies (IYL2015) provided a worldwide stimulus to optical sciences and associated technologies. The year-long celebrations of light were a good occasion to revisit the history of optics. The year 2015 had the millennium anniversary of the encyclopaedic seven volumes on optics by the legendary Arab scholar Ibn al-Haytham. Recent analysis by the mathematician and science historian Roshdi Rashed demonstrates that the law of refraction of light had been penned down in detail by Ibn Sahl, who was the teacher of Ibn al-Haytham. In the absence of this major historical discovery, we would have continued to attribute the law to the European Renaissance scientists. Here, we describe the Mediaeval Arab achievements in optics and its influence on the European sciences. Looking ahead, we also briefly examine the ways in which the light sciences can provide a platform for international science collaborations and revival of science in the Arab lands.

We address the theme of international science collaborations, in the light of examples of the institutions from Europe, which were built after the Second World War. These international institutions devoted to science played a pivotal role in the rebuilding of Europe devastated by the wars. CERN, the European Laboratory for Particle Physics in Geneva, Switzerland is a prime example of international collaboration, which now has 21 member countries. International science collaborations served as vehicles of communication and peace during the cold war. Synchrotron-light for Experimental Science and Applications in the Middle East (SESAME) in Jordan is modelled after CERN. Both CERN and SESAME are under the auspices of UNESCO. The SESAME facility is sure to advance science and promote cooperation in the region and beyond. The article discusses the need to create international science centres in Arab countries. The proposed centres can be modelled after the European institutions keeping in mind the local perspective. Such centres, when created, shall lead to the renaissance of sciences in the region.

An exact treatment of beam optics, starting ab initio from the Maxwell's equations is presented. The starting point of this approach is a matrix representation of the Maxwell's equation in a medium with varying permittivity and permeability. Formal expressions are obtained for the paraxial and leading order aberrating Hamiltonians, without making any assumptions on the form of the varying refractive index. We derive the wavelength-dependent contributions at each order, starting with the lowest-order paraxial Hamiltonian. To illustrate the general theory, we consider the computations of the transfer maps for an axially symmetric graded-index medium. For this system, in the traditional approaches, one gets only six aberrations. In our formalism, we get all the nine aberrations permitted by the axial symmetry. The six aberrations coefficients of the traditional approaches get modified by the wavelength-dependent contributions and the remaining three are pure wavelength-dependent. It is very interesting to note that apart from the wavelength-dependent modifications of the aberrations, this approach also gives rise to the image rotation. The present study is the generalization of the traditional and non-traditional prescription of Helmholtz optics. In the low wavelength limit our formalism reproduces the Lie algebraic formalism of optics. The present study further strengthens the close analogies between the various prescriptions of light optics and charged-particle optics. The new formalism presented here, provides a natural framework to study beam-optics and polarization in a unified manner.

A unified formalism of light beam optics and light polarization is presented. The starting point of our formalism is an exact matrix-representation of Maxwell's equations in an inhomogeneous medium, which is presented in detail. The beam-optical Hamiltonians are derived without any specifications on the form of the varying refractive index. The new formalism generalizes the traditional and non-traditional prescriptions of Helmholtz optics. As for the light polarization, the elegant Mukunda–Simon–Sudarshan rule for transition from scalar to vector wave optics is obtained as the paraxial limit of the general formalism presented here. The new formalism is a suitable candidate to extend traditional theory of polarization beyond the paraxial approximation. The unified formalism of light beam-optics and light polarization further strengthens the Hamilton's optical-mechanical analogy, particularly in the wavelength-dependent regime.

Given a bunch of equal resistors, what are the values of the equivalent resistances which one can obtain? Is there a formula? In this note, we address such questions analytically and establish strict lower and upper bounds for the order of the set of equivalent resistances using series and parallel connections. The approach of the Farey sequences is shown to be applicable to resistor networks involving bridge and non-planar circuits. For equal resistors, the order of the set is bounded from below by and from above by .

Objectives: Numerical calculus can be done using calculators, readymade packages or programming languages. The spreadsheets are a middle course between the programming languages and the readymade packages. Methods/ Statistical Analysis: Spreadsheet programs serve as a powerful tool for graphical analysis and programming. Findings: We shall describe how the ubiquitous Microsoft EXCEL can be used to do numerical calculus, with appropriate examples. Use of EXCEL does not require any prior knowledge of computer programming are straightforward to use. Applications/ Improvements: We apply the Microsoft EXCEL to the do numerical integration and to obtain numerical solutions of equations using the Newton-Raphson method.

It is straightforward to construct the set of equivalent resistance for circuits constructed from a bunch of four or five equal resistors. But as the bunch size increases it becomes difficult to find the order of the set of equivalent resistances. Even the computer programs runs out of memory. Here we present an analytical result using simple mathematical machinery. The size of the set is shown to be less than 2.618n.

The order of the set of equivalent resistances, A(n) of n equal resistors combined in series and in parallel has been traditionally addressed computationally, for n up to 22. For larger n there have been constraints of computer memory. Here, we present an analytical approach using the Farey sequence with Fibonacci numbers as its argument. The approximate formula, A(n) ~ 2.55^n, obtained from the computational data up to n = 22 is consistent with the strict upper bound, A(n) ~ 2.618^n, presented here. It is further shown that the Farey sequence approach, developed for the A(n) is applicable to configurations other than the series and/or parallel, namely the bridge circuits and non-planar circuits. Expressions describing set theoretic relations among the sets A(n) are presented in detail. For completeness, programs to generate the various integer sequences occurring in this study, using the symbolic computer language MATHEMATCA, are also presented.

The King Faisal Foundation in Riyadh, Saudi Arabia has awarded the 2018 King Faisal International Prize (KFIP) for Medicine (topic: Immunotherapy for Cancer) to James Patrick Allison of USA, for his outstanding contribution to the development of the field of immunotherapy of cancer. The Science prize in the field of mathematics has been awarded to John Macleod Ball of the UK for his fundamental contributions to nonlinear partial differential equations, calculus of variations, and their applications to materials science and liquid crystals. The Prize comprises a 24-carat gold medal weighing 200 g; and a cash award of 750 thousand Saudi Riyal (USD 200,000).

I read your Japan special report with a keen interest. My first visit to Japan was in March 1994 to attend the JSPS-KEK International Spring School: High Energy Ion Beams – Novel Beam Techniques and their Applications.

Matrix representations of Maxwell's equations have a striking resemblance to the Dirac equation. We exploit this resemblance to build a beam optics formalism from an exact eight-dimensional matrix representation of the Maxwell equations. The Foldy–Wouthuysen iterative diagonalization technique is employed to obtain a Hamiltonian description for a system with varying refractive index. The beam-optical Hamiltonian is shown to have a wavelength-dependent part, resulting in the wavelength-dependent modifications of light beam optics. The present study is the generalization of the traditional and non-traditional prescription of Helmholtz optics.

The classical treatments have been successful in designing numerous charged-particle devices. It is natural to develop a quantum prescription, since all systems are fundamentally quantum mechanical in nature. The quantum theory leads to new insights accompanied with wavelength-dependent contributions. The action of a magnetic quadrupole is derived from the Dirac equation.

We report here synthesis and subsequent nitrogen ion implantation of indium oxide (In2O3) thin films. The films were implanted with 25keV N+ beam for different ion doses between 3E15 to 1E16 ions/cm2. The resulting changes in structural and optical properties were investigated using XRD, SEM-EDAX and UV-Vis Spectrometry. XRD studies reveal decrease in crystallite size from 20.06 to 12.42 nm with increase in ion dose. SEM micrographs show an increase in the grain size from 0.8 to 1.35 µm with increase in ion dose because of the agglomeration of the grains. Also, from EDAX data on pristine and N-implanted thin films the presence of indium and oxygen without any traces of impurity elements could be seen. However, at lower ion doses such as 3E15 and 5E15 ions/cm2, no evidence of the presence of nitrogen ion was seen. However, for the ion dose of 1E16 ions/cm2, evidence of presence of nitrogen can be seen in the EDAX data. Band gap calculations reveal a decrease in band gap from 3.54 to 3.38eV with increasin...

The Helmholtz wave equation is linearized using the Feshbach–Villars procedure used for linearizing the Klein–Gordon equation, based on the close algebraic analogy between the Helmholtz equation and the Klein–Gordon equation for a spin-0 particle. The Foldy–Wouthuysen iterative diagonalization technique is then applied to the linearized Helmholtz equation to obtain a Hamiltonian description for a system with varying refractive index. The Hamiltonian has a wavelength-dependent part absent in the traditional descriptions. Besides reproducing all the traditional quasi-paraxial terms, our method leads to additional contributions dependent on the wavelength. Applied to the axially symmetric graded-index fiber, this method results in wavelength-dependent modifications of the paraxial behavior and the aberration coefficients to all orders. Explicit expression for the modified aberration coefficients to the third order are presented. Sixth- and eighth-order Hamiltonians are also presented.

Over three years ago, Germany decided to gift the BESSY-I, a 800MeV synchrotron, fully functioning since 1982 in Berlin, to the region of Middle East. The Middle East Synchrotron better known by the acronym SESAME (Synchrotron-light for Experimental Science and Applications in the Middle East) will be the upgraded reincarnation of BESSY-I. SESAME, the first synchrotron in the Middle East, is envisioned as a seed for a regional international research center, open to scientists in the region and beyond. The founders of the SESAME Project see a facility similar in purpose to the European Laboratory for Particle Physics (CERN) in Geneva, which brought together numerous scientists from countries that had fought each other during the two World Wars. Scientific collaboration beyond the national boundaries has been a force for peace in the cold war. An account of events leading to these very significant developments is presented.

In this article we report the phenomenon of motion of salt along the walls of the container containing the salt solution. When a table salt solution is left exposed to the atmosphere, the water naturally evaporates in several days. It is observed that as the water evaporates a thin layer of salt starts to form on the inner walls of the glass. This layer, from the inception, tends to form above the original level of the solution. Surprisingly, the layer spreads to the top of the glass and continues on to the outer wall of the glass. This surprising behaviour of the salt motion is described in detail along with the photographs.